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Full-Order Safety for Nonlinear Reduced-Order MPC

Luis Pabon, Johannes Köhler, Revant Kasichainula, Marco Pavone

arXiv:2610.19250v1cs.ROcs.SYeess.SYmath.OC

TL;DR

ROMs make predictive control of high-dimensional nonlinear systems tractable, but modeling error can violate constraints on the real system. CoRE-MPC plans with a nonlinear ROM and propagates two radii to bound coupled prediction errors while guaranteeing full-order constraint satisfaction and recursive feasibility. In the reported examples, it is 13 to 61 times faster than full-order robust MPC at 27 to 35% higher closed-loop cost.

  • Problem

    ROM predictions can differ from the real system enough to violate its constraints, while existing nonlinear ROM-based schemes can be overly conservative, infeasible, or restricted to invariant manifolds.

    Source: §I · p. 1

  • Method

    CoRE-MPC plans with a nonlinear ROM and propagates two coupled error radii alongside it to bound prediction error and guarantee full-order constraints without online full-order evaluation.

    Source: §I · p. 2

  • Results

    13 to 61 times faster than full-order robust MPC at 27 to 35% higher closed-loop cost, CoRE-MPC keeps all 1,628 soft-robot nodes clear of the wall with median solve time below 1 ms.

    Source: §I · p. 2

  • Takeaways & Limitations

    CoRE-MPC applies to a general class of nonlinear ROMs, including non-invariant approximation manifolds and ROMs learned from data, and supports real-time soft-robot collision avoidance where full-order robust MPC is prohibitive.

    Source: §I · p. 2, §VI · p. 8

Abstract

from arXiv · show

Reduced-order models (ROMs) make model predictive control (MPC) of high-dimensional nonlinear systems tractable, but the resulting modeling error can lead to constraint violations on the real system. We introduce CoRE-MPC, a robust MPC formulation that plans with a nonlinear ROM and guarantees recursive feasibility and constraint satisfaction for the full-order system. It can be applied with a broad class of nonlinear model reduction techniques and the resulting MPC does not require evaluating the full-order model online, thus ensuring computational efficiency. The key insight is in how the ROM affects the dynamics of the prediction error. We account for this with two error radii that are propagated alongside the ROM and provide a guaranteed bound on the prediction error. The approach is validated in simulation on an 80-state oscillator chain, a Burgers equation, and a 9,768-state finite-element soft robot, where a two-state ROM with data-driven bounds keeps all 1,628 nodes clear of a wall at a median solve time below 1 ms for a 20 ms control period. Across these examples, CoRE-MPC is 13 to 61 times faster than full-order robust MPC at only 27 to 35% higher closed-loop cost, whereas existing certified schemes are either infeasible or at least twice as conservative. Code: [released upon publication].

I. INTRODUCTION

ROMs make predictive control tractable for high-dimensional nonlinear systems, but their predictions can diverge from the real system and fail to ensure its constraints. CoRE-MPC addresses this by bounding coupled prediction errors while planning with a nonlinear ROM.

I. INTRODUCTION

  • ROM mismatch can make a predicted collision-free trajectory collide on the physical system, and hand-tuned margins or larger ROMs trade performance and computation for safety without guaranteeing it.The prediction error depends on the full state, including unmeasured components whose dynamics feed back into the reduced prediction.Source: §I · p. 1
  • Existing nonlinear robust MPC can be overly conservative or infeasible on ROMs, while the only cited scheme certifying full-order constraints requires an invariant manifold near an equilibrium.CoRE-MPC is presented for a general class of nonlinear ROMs.Source: §I · pp. 1–2
  • CoRE-MPC bounds the prediction error inside and outside the reduced coordinates separately, accounting for their coupling.The reduced-coordinate error can be reset at each update, whereas the error outside those coordinates cannot.Source: §I · p. 2
  • A coupled reduction-error certificate bounds prediction error for general ROMs using a two-dimensional comparison system.Online, two error radii are propagated with the ROM at a cost comparable to nominal ROM-MPC with two additional states.Source: §I · p. 2
  • On the oscillator chain, CoRE-MPC is 13 to 61 times faster than full-order robust MPC at 27 to 35% higher closed-loop cost; on the soft robot, it keeps all 1,628 nodes clear of the wall with median solve time below 1 ms.Source: §I · p. 2

II. PROBLEM FORMULATION

The problem is to control a full-order system using a reduced-order model without evaluating the full model online. The controller must satisfy the real system’s constraints under admissible disturbances while minimizing task cost.

II. PROBLEM FORMULATION

  • Input constraints can represent actuator limits, while state constraints can enforce collision avoidance and workspace bounds over the whole soft robot.The input set is compact and the constraint function is C1.Source: §II · p. 2
  • The ROM uses a reduced state and locally Lipschitz dynamics, with a C2 lifting that reconstructs the full state.The lifting’s image is the approximation manifold, and P extracts reduced coordinates while acting as a left inverse of the lifting.Source: §II · p. 2
  • The formulation includes linear projections and nonlinear liftings, with ROM dynamics either projected from the full-order model or learned from data.The results require neither an invariant approximation manifold nor a stable ROM, although both affect error-bound growth.Source: §II · p. 2
  • At update times, the controller measures only reduced coordinates and assumes a known bound on the initial distance from the system state to the approximation manifold.The initial bound is zero if the system starts on the manifold.Source: §II · p. 2
  • The controller must plan with the ROM without evaluating the full-order model online, guarantee full-order constraints under every admissible disturbance, and minimize task cost.The full-order model has state x, input u, C1 dynamics, and a bounded piecewise-continuous disturbance.Source: §II · p. 2

III. COUPLED REDUCTION-ERROR CERTIFICATES

The prediction error is bounded using separate radii for its two components, whose growth and coupling rates are bounded to construct the certificate.

III. COUPLED REDUCTION-ERROR CERTIFICATES

  • The radii ρ = (ρ∥, ρ⊥) bound the two components of the prediction error x − Φ(z).Their growth and coupling rates are bounded, and the bounds are constructed offline.Source: §III · p. 2

A. Tangential and Transverse Error Dynamics

The prediction error is split into tangential error in reduced coordinates and transverse error away from the approximation manifold. Their dynamics are coupled, and only the tangential component is determined by the reduced-coordinate measurement.

A. Tangential and Transverse Error Dynamics

  • The tangential error measures prediction error in reduced coordinates, while the transverse error measures departure from the approximation manifold.The projection Φ◦P defines the split.Source: A. Tangential and Transverse Error… · p. 2
  • The reduced-coordinate measurement determines the tangential error but leaves the high-dimensional transverse error unmeasured.The transverse error has its own dynamics.Source: A. Tangential and Transverse Error… · p. 2
  • The exact error dynamics are coupled through the full-order flow, with Q projecting onto ker P along the manifold’s tangent space.Q is defined as I − DΦ(ζ)P.Source: A. Tangential and Transverse Error… · pp. 2–3
  • The transverse residual is the rate at which the full-order flow leaves the manifold, while the tangential residual measures on-manifold flow mismatch.The tangential residual vanishes when the ROM dynamics are the projected full-order flow.Source: A. Tangential and Transverse Error… · p. 3

B. Comparison System for the Error Norms

The comparison system bounds tangential and transverse prediction errors separately while accounting for their coupling. Under its stated conditions, the two propagated radii contain the corresponding errors.

B. Comparison System for the Error Norms

  • The fixed error domain trades domain of validity against conservatism, while radius-dependent bounds can remove the fixed-radius condition.The radius-dependent result requires continuous, locally Lipschitz bounds that are componentwise nondecreasing in the radii.Source: B. Comparison System for the Error Norms · p. 3
  • The error bounds separate growth rates for each error component, cross-coupling between components, and forcing that persists at zero error.The diagonal rates describe component-wise growth or contraction; the off-diagonal rates capture coupling, and b bounds residual and disturbance forcing.Source: B. Comparison System for the Error Norms · p. 3
  • The certificate propagates two radii through a comparison system to bound tangential and transverse errors.Theorem 1 guarantees both error norms stay within their radii when the initial radii cover the errors and the radii remain below the prescribed bound.Source: B. Comparison System for the Error Norms · p. 3
  • The bounds can be obtained from Jacobian block norms and logarithmic norms over the chosen error set.The Jacobian blocks with respect to tangential error are evaluated at zero transverse error; blocks with respect to transverse error act only on ker P.Source: B. Comparison System for the Error Norms · p. 4
  • The radii remain ultimately bounded under a constant Hurwitz bound, but re-centering can also permit use when that bound is not Hurwitz.For invariant manifolds the matrix is upper triangular; for general POD projections, the more general coupled structure is required.Source: B. Comparison System for the Error Norms · p. 4

IV. CORE-MPC: PREDICTIVE CONTROL WITH COUPLED ERROR CERTIFICATES

CoRE-MPC tightens constraints using the propagated error radii and optimizes over the ROM augmented by those radii. With the stated terminal ingredients and feasibility assumptions, it guarantees recursive feasibility and full-order constraint satisfaction without evaluating the full-order vector field online.

Constraint tightening

  • Constraint tightening bounds each full-order constraint over the error tube and enforces the resulting bound in place of the original constraint.For affine constraints, the transverse contribution involving ker P is exact.Source: §IV · p. 4

Online optimization

  • CoRE-MPC augments the ROM with two error radii, making its computational demand comparable to nominal MPC for an (r+2)-dimensional system.The full-order vector field is never evaluated online; state- and input-dependent bounds allow the optimizer to trade task performance against tube growth.Source: §IV · pp. 4–5

Terminal ingredients

  • The terminal set must tolerate radius growth, since stability of the ROM alone does not ensure viability of the augmented state.A constant Hurwitz bound can provide an invariant radius box for a product terminal set.Source: §IV · p. 5

Closed-loop guarantees

  • Feasibility at the initial update, valid bounds, and the terminal assumptions yield recursive feasibility and full-order constraint satisfaction under CoRE-MPC.The certificate contains the errors over each sampling interval, and tightened constraints then imply the original state constraints.Source: §IV · p. 5
  • The terminal ingredients support a shifted-plan argument that carries feasibility from one update to the next.The shifted plan is extended with the terminal law while the augmented state remains in the invariant terminal set.Source: §IV · p. 5

V. EXPERIMENTS

The experiments evaluate CoRE-MPC on an oscillator chain, a Burgers equation, and a finite-element soft robot, comparing it with robust, nominal, and simplified robust controllers. The examples test invariant-manifold and non-invariant reductions as well as whole-body constraints.

Experimental systems

  • The oscillator chain tests the tradeoff among full-order safety, closed-loop cost, and computation when the invariant-manifold assumption holds.The Burgers test instead uses a POD reduction whose manifold is not invariant and closed-form bounds valid for every state and radius.Source: §V · p. 5
  • The three tests cover an oscillator chain, a Burgers equation, and a 9,768-state soft robot with a two-state ROM and whole-body constraints.The soft-robot test uses bounds fitted from data and checks all 1,628 nodes.Source: §V · p. 5

Controllers

  • Comparisons include CoRE-MPC, constant-bound CoRE, nominal ROM-MPC, full-order robust MPC, RN-ROMPC, and linear ROMPC.The full-order robust controller is the benchmark for cost and computation time.Source: §V · p. 5

Implementation and evaluation

  • Controller solve times were measured with a common SQP setup, and all shown controllers converged at every update.The evaluation used acados with HPIPM on an Apple M3 Max; solve times exclude fallback to a shifted previous plan.Source: §V · p. 5

Experimental measurements

  • The oscillator-chain figure compares maximum normalized displacement, median cost relative to FOM-MPC, and controller solve times.The Burgers figure also shows the field, maximum normalized field value, predicted radii against realized errors, and solve times.Source: Figure 2 · p. 6, Figure 3 · p. 6

A. Oscillator Chain

Across 48 oscillator-chain cases, robust controllers satisfy the displacement constraints while nominal ROM-MPC violates them in some cases. CoRE-MPC offers the lowest excess cost among the robust controllers and substantially faster solves than full-order robust MPC.

Results

  • 61 times faster than FOM-MPC, CoRE-MPC has 35% excess closed-loop cost while satisfying constraints in all 48 cases.Nominal ROM-MPC violates the displacement limit in 23 of 48 cases, by up to 4% of qmax; the figure compares median cost and solve-time distributions.Source: A. Oscillator Chain · p. 6
  • 149% excess closed-loop cost for RN-ROMPC and 49% for Constant CoRE exceed CoRE-MPC's 35%.Replacing state- and input-dependent bounds by constants makes the tubes grow at the worst-case rate along the plan.Source: A. Oscillator Chain · p. 6
  • Linear ROMPC is infeasible in all 48 cases because bounding the nonlinear spring force as a disturbance is too conservative.The oscillator has 80 states, a two-state spectral-submanifold ROM, and constraints on every mass displacement.Source: A. Oscillator Chain · p. 6
  • CoRE-MPC's 95th-percentile solve time is 10.6 ms against the 100 ms sampling time.Its solve time is 2.9 times that of nominal ROM-MPC.Source: A. Oscillator Chain · p. 6

B. Burgers Equation

In the Burgers tests, CoRE-MPC satisfies the constraints in all 24 cases despite nominal ROM-MPC violations, with a 27% higher closed-loop cost than FOM-MPC and 13 times faster solves. Its coupled error certificate remains valid and tight even though the approximation manifold is non-invariant and the comparison matrix is not Hurwitz.

B. Burgers Equation

  • The six-mode POD–Galerkin ROM captures 99.998% of snapshot energy but has a non-invariant approximation manifold; 24 disturbance phases define the test cases.The reference peaks 54% above the field limit, and the localized sinusoidal disturbance acts on the field.Source: B. Burgers Equation · p. 7
  • 27% higher closed-loop cost than FOM-MPC and 13 times faster solves accompany constraint satisfaction in all 24 cases, while nominal ROM-MPC violates the limit in every case.CoRE-MPC's 95th-percentile solve time is 12.3 ms against a 100 ms sampling time.Source: B. Burgers Equation · p. 7
  • Nonzero couplings and transverse residual make the coupled comparison system necessary, although its matrix M is not Hurwitz.Free initialization lets the tangential radius restart at zero after the first update, while the carried transverse radius stays bounded.Source: B. Burgers Equation · p. 7
  • Every predicted tube contains the realized errors, which reach 63% of the transverse radius and 98% of the tangential radius across the cases.Along evaluated trajectories, coupling bounds exceed exact Jacobian block norms by at most factors of 1.15 and 1.38.Source: B. Burgers Equation · p. 7

C. Finite-Element Soft Robot

In the finite-element soft-robot task, all three robust controllers keep every node clear of the wall, unlike nominal ROM-MPC. CoRE-MPC combines lower tracking error than the other robust controllers with sub-period solve times, and held-out runs maintain clearance while the error radii contain realized errors.

C. Finite-Element Soft Robot

  • The 9,768-state robot uses a two-state spectral-submanifold ROM, with data-fitted error bounds from ten earlier rollouts and a 60 ms horizon at 20 ms sampling.Only nodes with the largest predicted wall envelopes enter optimization, but every plan is checked against all 1,628 nodes.Source: C. Finite-Element Soft Robot · p. 7
  • All three robust controllers keep every node clear, with minimum clearances of 3.7 mm for CoRE-MPC, 5.4 mm for Constant CoRE, and 0.5 mm for RN-ROMPC, while nominal ROM-MPC penetrates by 13 mm.CoRE-MPC tracks with 29% less error than Constant CoRE and 38% less than RN-ROMPC.Source: C. Finite-Element Soft Robot · p. 7
  • CoRE-MPC solves in 0.9 ms at the median against 0.55 ms for nominal ROM-MPC, never exceeds 5 ms, and has two RN-ROMPC updates exceed the 20 ms period.Full-order MPC is excluded because one FEM horizon prediction takes 0.56 s, or 28 sampling periods.Source: C. Finite-Element Soft Robot · p. 8
  • In all 14 held-out runs, every node stays clear and both radii contain realized errors at every update, reaching at most 90% of the tangential and 77% of the transverse radius.With the larger calibration set, CoRE-MPC's RMS tracking error is 17.4 mm, 20% less than Constant CoRE and 50% less than RN-ROMPC.Source: C. Finite-Element Soft Robot · p. 8

VI. CONCLUSION

CoRE-MPC propagates separate tangential and transverse error radii alongside nonlinear ROMs to enforce full-order constraints, including for non-invariant manifolds and data-learned ROMs. The numerical studies show lower excess cost than existing certified schemes where applicable and real-time whole-body collision avoidance for a soft robot where full-order robust MPC is prohibitive.

VI. CONCLUSION

  • CoRE-MPC enforces full-order constraints for a general class of nonlinear ROMs, including non-invariant manifolds and ROMs learned from data.Because reduced-coordinate error can be reset but transverse error cannot, the method bounds both separately and propagates two radii with the ROM at the cost of two extra states.Source: §VI · p. 8
  • The numerical studies show a fraction of the excess cost of existing certified schemes where they apply, and real-time whole-body collision avoidance for a soft robot where full-order robust MPC is prohibitive.Open issues are certification of comparably tight finite-element-scale bounds, currently fitted to data, and additional sensing of the open-loop propagated transverse error.Source: §VI · p. 8

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