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Taylor-Informed Indirect Adaptive Predictive Control Using Jacobian-Frozen Affine Predictors

Tam W. Nguyen

arXiv:2609.03348v1eess.SYcs.SY

TL;DR

Generic nonlinear dictionaries may be only indirectly connected to the plant, while Taylor features offer a local representation when structural knowledge is partial. The paper identifies Taylor-map coefficients online and uses a Jacobian-frozen affine predictor for MPC; simulations report improved tracking for higher Taylor degrees with similar control variation.

  • Problem

    Generic nonlinear feature dictionaries can be indirectly connected to the plant, while applications may provide only partial structural knowledge for choosing a basis.

    Source: §I · p. 1

  • Method

    The framework identifies a Taylor-structured polynomial map online and freezes its Jacobian at the current operating point to form an affine MPC predictor.

    Source: §I · p. 1

  • Results

    Higher-order models D ∈{3, 5, 7} show lower overall MAE than D = 1, with similar total input variation.

    Source: §IX · p. 6

  • Takeaways & Limitations

    Joint-odd symmetry exactly removes even-total-degree monomials, while additional forward-Euler-informed pruning is deliberate and does not mean omitted monomials vanish in the RK4 sampled flow.

    Source: §IX · p. 6

    The paper asserts no recursive-feasibility or closed-loop stability guarantee because the identified model and Jacobian-frozen predictor vary online.

    Source: §VI · p. 3

Abstract

from arXiv · show

This paper develops a Taylor-informed indirect adaptive predictive control framework for nonlinear sampled-data systems using Jacobian-frozen affine predictors. A finite Taylor expansion approximates the sampled nonlinear dynamics, and recursive least squares (RLS) identifies its polynomial coefficients online. At each sampling instant, the Jacobian of the identified map is evaluated at the current operating point and frozen over the prediction horizon, yielding an affine predictor for model predictive control. In contrast to generic nonlinear feature dictionaries, the implemented polynomial dictionary is a forward-Euler/Taylor-structure-informed reduced dictionary. Exact joint-odd symmetry eliminates even-total-degree monomials, whereas additional forward-Euler-informed pruning constitutes a deliberate model reduction. Numerical simulations on an unstable nonlinear benchmark compare different Taylor degrees. The results show that higher-order models improve tracking accuracy as the operating point moves farther from the expansion point while maintaining comparable control effort. The complete MATLAB implementation is publicly available to facilitate reproducibility.

I. INTRODUCTION

The paper replaces generic nonlinear dictionaries with a reduced Taylor-structure-informed feature set, then uses online identification and a locally affine predictor for MPC. It frames this approach for sampled nonlinear dynamics around a selected equilibrium.

I. INTRODUCTION

  • Generic nonlinear dictionaries can improve approximation while keeping coefficients linear in the parameters, but their connection to the plant may be indirect.A Taylor dictionary offers a common local representation when only partial structural knowledge is available; increasing its degree allows richer local behavior through coefficient adaptation.Source: §I · p. 1
  • The proposed reduced feature set follows the Taylor structure of sampled dynamics and forward-Euler information rather than a generic nonlinear dictionary.Joint-odd symmetry justifies some eliminations, while additional forward-Euler-informed omissions are deliberate model reduction and do not imply those monomials vanish in the sampled flow.Source: §I · p. 1
  • Online RLS identifies Taylor coefficients, and the identified map’s Jacobian is frozen at the current operating point to provide an affine predictor for conventional MPC.This avoids repeated nonlinear propagation while retaining adaptation through the identified coefficients.Source: §I · p. 1

II. SYSTEM DESCRIPTION

  • The system description shifts the sampled-data dynamics around an equilibrium pair and assumes state feedback, a well-defined local map, and sufficient smoothness for a finite Taylor approximation.The sampled flow map is associated with zero-order-hold dynamics.Source: §II · pp. 1–2

III. TAYLOR APPROXIMATION OF THE SAMPLED DYNAMICS

The section approximates the sampled nonlinear map locally with a finite Taylor polynomial and expresses it as a linear function of polynomial features. This form makes the coefficients amenable to identification.

III. TAYLOR APPROXIMATION OF THE SAMPLED DYNAMICS

  • The sampled map admits a finite Taylor approximation about the origin under the stated smoothness assumption, with a remainder term.Each F_d collects homogeneous polynomial terms of total degree d.Source: §III · p. 2
  • The order-D regressor contains pure and crossed monomials of total degree 1 through D, formed from the components of z.The full Taylor index set specifies which monomials enter the polynomial regressor.Source: §III · p. 2
  • The sampled map is represented as Θϕ_D(z) plus the Taylor remainder, making the approximation linear in coefficient matrix Θ.Neglecting the remainder reduces identification to estimating Θ.Source: §III · p. 2

IV. RECURSIVE TAYLOR COEFFICIENT IDENTIFICATION

The method estimates the Taylor-map coefficients online from one-step transitions using RLS. It then uses the most recently identified nonlinear map to construct a Jacobian-frozen affine predictor.

IV. RECURSIVE TAYLOR COEFFICIENT IDENTIFICATION

  • After neglecting the Taylor remainder, identifying the sampled dynamics reduces to estimating the Taylor coefficient matrix.The polynomial model is linear in its coefficients, so the one-step prediction error feeds a recursive least-squares update.Source: §IV · p. 2
  • In the numerical study, RLS processes one-step transitions from the zero-order-hold RK4 plant simulation; forward Euler does not propagate the simulated plant.The posterior coefficient estimate is available after the next state is measured.Source: §IV · p. 2

V. JACOBIAN-FROZEN AFFINE PREDICTOR

  • The identified map remains nonlinear in its inputs but linear in its estimated coefficients, and other RLS variants can preserve the Jacobian-frozen construction.Exact finite-horizon propagation would require nonlinear predictive control, motivating a local affine predictor formed by freezing the map’s Jacobian.Source: §IV · p. 2, §V · p. 2

A. Operating Point

The predictor evaluates the identified map’s Jacobian at the current operating point and holds the resulting affine coefficients fixed across the horizon. At first Taylor order, the propagation reduces to linear PCAC.

A. Operating Point

  • The operating point uses the measured shifted state and currently applied shifted input, while the Taylor approximation remains centered at the equilibrium.The operating point is used to evaluate the Jacobian, not to recenter the Taylor approximation.Source: A. Operating Point · pp. 2–3

B. Jacobian Evaluation

  • The Jacobian partitions into state and input matrices, and the local first-order approximation yields an affine predictor whose coefficients remain fixed over the horizon.The affine offset is computed from the identified map evaluated at the operating point.Source: B. Jacobian Evaluation · p. 3

C. Reduction to Linear PCAC

  • For D = 1, the Jacobian-frozen predictor reduces to the linear PCAC propagation model.In this case the Jacobian is independent of the operating point and the affine offset is zero.Source: C. Reduction to Linear PCAC · p. 3

D. Elimination of Factor-Freezing Ambiguity

Jacobian freezing avoids arbitrary choices for mixed nonlinear terms while preserving the identified map’s local value and Jacobian. The predictive-control synthesis uses this frozen affine model in a horizon-based quadratic program, but does not assert recursive feasibility or closed-loop stability.

D. Elimination of Factor-Freezing Ambiguity

  • Freezing the Jacobian avoids arbitrary choices for approximating mixed nonlinear terms.For the cross term uy, the approximation uses both operating-point factors and matches the map’s value and Jacobian there.Source: D. Elimination of Factor-Freezing… · p. 3
  • The cross-term approximation uy ≈ ūy + ȳu − ūȳ matches both the value and Jacobian at the operating point.It does so without choosing whether to freeze u or y.Source: D. Elimination of Factor-Freezing… · p. 3

VI. PREDICTIVE CONTROL SYNTHESIS

  • The frozen affine predictor propagates states across the horizon, with the current shifted input known and later inputs optimized as decision variables.The indexing convention applies the first optimized input at step k + 1.Source: §VI · p. 3
  • The synthesis is a convex quadratic program, but soft state constraints can be violated and no recursive-feasibility or closed-loop-stability guarantee is asserted.The slack variables prevent infeasibility caused solely by state constraints while permitting violation of nominal state constraints.Source: §VI · p. 3

VII. CASE STUDY: UNSTABLE TRIGONOMETRIC PLANT

The case study uses an open-loop-unstable trigonometric plant and estimates its sampled transitions from an RK4 simulation. Joint sign-reversal symmetry also constrains the sampled-flow and RK4 Taylor expansions.

A. Plant and Sampled Dynamics

  • The benchmark is an open-loop-unstable trigonometric plant whose equilibrium linearization is controllable.The plant is studied around the equilibrium (0, 0).Source: A. Plant and Sampled Dynamics · pp. 3–4
  • With zero-order hold and Ts = 0.05 s, RK4 propagates the plant and supplies the one-step transitions identified by RLS; forward Euler is only an analytical reference.The simulated input is held constant over each sampling interval.Source: A. Plant and Sampled Dynamics · p. 4
  • Joint sign-reversal symmetry means the sampled flow and implemented RK4 map have no even-total-degree terms in their Taylor expansions about the origin.The identity holds exactly for the sampled flow and for the RK4 map because each RK4 stage preserves the sign reversal.Source: A. Plant and Sampled Dynamics · p. 4

B. Taylor Structure and Coefficient Pruning

The implemented dictionary combines exact symmetry-based elimination with deliberate forward-Euler-informed pruning. This reduces the number of estimated coefficients, while the affine term is introduced later through Jacobian freezing.

B. Taylor Structure and Coefficient Pruning

  • The degree-five forward-Euler expansion provides an analytical reference for the polynomial terms, not the map propagated in the numerical study.The simulation’s one-step transitions are identified from the RK4-propagated plant.Source: B. Taylor Structure and Coefficient… · p. 4
  • Joint-odd symmetry removes even-total-degree terms exactly, while additional forward-Euler-informed pruning is deliberate and does not imply omitted odd-degree terms vanish in the sampled flow.The implemented feature set is a forward-Euler/Taylor-structure-informed reduced dictionary.Source: B. Taylor Structure and Coefficient… · p. 4
  • The reduced dictionary cuts estimated coefficients from 20 to 9 for D = 5 and from 35 to 14 for D = 7.Table I compares coefficient counts in the full and implemented reduced dictionaries.Source: B. Taylor Structure and Coefficient… · p. 4, Table I · p. 4
  • No constant monomial is estimated; the affine term arises subsequently from Jacobian freezing.RLS estimates the retained coefficients online rather than using the analytical forward-Euler coefficients.Source: B. Taylor Structure and Coefficient… · p. 4

C. Identified Map and Jacobian-Frozen Predictor

The identified sampled map is updated online with RLS, then used to form a Jacobian-frozen predictor whose coefficients are evaluated at the current operating point. The predictor advances states through an affine recursion over the horizon.

C. Identified Map and Jacobian-Frozen Predictor

VIII. NUMERICAL EVALUATION

The numerical evaluation compares Taylor degrees in an RK4-simulated nonlinear plant, using a shared initialization and controller configuration. The setup progressively increases reference amplitudes to excite higher-order nonlinearities.

VIII. NUMERICAL EVALUATION

  • All Taylor degrees D ∈ {1, 3, 5, 7} use the same initialization data, constraints, prediction horizon, and control weights in the RK4 evaluation.RLS identifies the one-step transitions generated by the sampled plant simulation.Source: §VIII · p. 5
  • The complete MATLAB implementation used in the study is publicly available.Source: §VIII · p. 5

A. Simulation Setup

  • The controller starts after multilevel-input RLS initialization; the state bound is softened and the input bound is hard.The initialization levels excite the retained odd monomials.Source: A. Simulation Setup · p. 5
  • Figures 1 and 2 display state trajectories and logarithmic absolute tracking errors, respectively, including segment transients in the error plot.Source: Figure 1 · p. 5, Figure 2 · p. 5, A. Simulation Setup · p. 5
  • Reference amplitudes 0.25, 0.55, 0.80, 0.90, and 0 are each held for 300 samples, progressively exciting higher-order nonlinearities.The state may exceed its nominal bound because that constraint is soft.Source: A. Simulation Setup · p. 5

B. Closed-Loop Results

Higher Taylor degrees reduce tracking error relative to first order as reference amplitude increases, with similar overall input variation. The section reports closed-loop tracking behavior, metric definitions, and boundaries of the model reduction and validation scope.

B. Closed-Loop Results

  • First- and fifth-order responses track the reference, with visible clipping near the softened state bound during larger-amplitude segments.The nonzero state violations mean exact state-constraint satisfaction is not claimed.Source: B. Closed-Loop Results · pp. 5–6
  • At larger amplitudes, the first-order model has consistently larger post-transient MAE than the third-, fifth-, and seventh-order models, which remain close; differences are negligible at small amplitudes.Figure 3 evaluates each amplitude segment after discarding its first 200 samples and averaging over the remaining 100.Source: Figure 3 · p. 6, B. Closed-Loop Results · p. 6

IX. CONCLUSION

  • Overall MAE is lower for D ∈{3, 5, 7} than for D = 1, with similar total input variation.Table III is identified as the closed-loop overall performance comparison across Taylor degrees.Source: Table III · p. 6, §IX · p. 6

B. Closed-Loop Results

  • Figure 2 displays absolute tracking errors, including segment transients, on a logarithmic vertical scale.The scale shows small within-segment errors and larger error peaks together.Source: B. Closed-Loop Results · p. 6

IX. CONCLUSION

  • Online RLS identifies the reduced sampled nonlinear map, whose Jacobian is evaluated at the current operating point and frozen over the prediction horizon to form an affine MPC predictor.The numerical study simulates the continuous-time plant under zero-order hold using RK4 with Ts = 0.05 s.Source: §IX · p. 6
  • Even-total-degree monomials are removed by exact joint-odd symmetry, while additional forward-Euler-informed pruning does not imply omitted monomials vanish in the RK4 sampled flow.The conclusion also reports maximum soft-state violations.Source: §IX · p. 6
  • Variable-rate forgetting, time-delay systems, and validation on real-world physical platforms are identified as future work.The paper proposes these extensions for investigation rather than reporting them as evaluated results.Source: §IX · p. 6
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