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Model-Based Galerkin Lifting with Exact LTI Decomposition for Guaranteed $\mathcal{H}_\infty$ Output Feedback Control of Nonlinear Systems

Burak Kurkcu, Christopher Phan, Aleksandar Zecevic, Maryam Khanbaghi

arXiv:2610.07299v1math.OCcs.SYeess.SY

TL;DR

Stabilizing a finite-dimensional approximation does not by itself guarantee stability of the original nonlinear system. This paper combines a model-based Galerkin realization with regional residual bounds in H∞ output-feedback design. Under a containment condition, it establishes closed-loop bounds and reports nominal tracking RMSE close to backstepping with a lower peak tracking error.

  • Problem

    Stabilizing an approximate lifted model does not by itself guarantee stability of the original nonlinear system, so approximation error must be accounted for in feedback design.

    Source: §I · p. 1

  • Method

    A model-based Galerkin lift preserves the actuator block and constant input matrix while retaining finite-order nonclosure as an exact additive residual for output-feedback synthesis.

    Source: §I · p. 2

  • Results

    The cart–pendulum example reports almost the same nominal tracking RMSE as full-state backstepping, with a lower peak tracking error, and conditional forward-completeness and boundedness guarantees.

    Source: §I · p. 2, §VII · p. 13

  • Takeaways & Limitations

    Retaining physical-state coordinates lets the regional certificates bound the original state and tracking error when the containment condition holds.

    Source: §I · p. 2

    The analysis assumes a known nonlinear vector field; extensions to identified models require accounting for model error to guarantee the true system.

    Source: §VII · p. 13

Abstract

from arXiv · show

In this paper, we present a model-based Galerkin framework for output-feedback control of nonlinear systems with known dynamics. With a suitable actuator augmentation, the finite-dimensional realization preserves the actuator dynamics and the constant input matrix. We retain finite-order nonclosure as an explicit additive residual. The LTI part and this residual describe the lifted nonlinear dynamics exactly. The decomposition does not require an invariant subspace of observables and also applies to non-control-affine systems. We use regional bounds on the closure residual, actuator-realization defect, and output-reconstruction error in the standard $\mathcal{H}_\infty$ design. The proposed controller is a linear dynamic system driven only by the measured tracking error and requires no online lifting. Retaining the state coordinates gives direct bounds on the original state. We use these bounds to derive an a priori containment condition. Under this condition, the nonlinear closed loop is forward complete, and the state and tracking error satisfy explicit transient bounds and are uniformly ultimately bounded. The cart-pendulum example illustrates the nonlinear closed-loop guarantees and compares the controller with full-state backstepping. Although the output-feedback controller uses only the measured tracking error, it achieves almost the same nominal tracking RMSE as backstepping, with a lower peak tracking error.

I. INTRODUCTION

The introduction motivates accounting for finite-dimensional approximation error during output-feedback synthesis, since stabilizing a lifted or linearized model alone does not guarantee stability of the nonlinear system. The paper develops a Galerkin realization and uses regional bounds to certify the original closed loop under a containment condition.

I. INTRODUCTION

  • Approximate-model stabilization alone does not guarantee stability of the original nonlinear system, and post-design region estimates do not incorporate regional approximation-error bounds into output-feedback synthesis.Linearization can offer local stability without specifying its guaranteed neighborhood.Source: §I · p. 1
  • The Galerkin construction preserves actuator dynamics and the constant input matrix while representing finite-order nonclosure as an exact additive residual, without requiring an invariant observable subspace or a control-affine plant.Retaining physical-state coordinates provides an exact state readout.Source: §I · p. 2
  • Regional residual bounds yield a standard H∞ output-feedback design and explicit state and tracking-error bounds; containment in the operating region ensures forward completeness and uniform ultimate boundedness.The containment condition can be checked using the physical-state image of the closed-loop ellipsoid or an auxiliary state bound.Source: §I · p. 2

II. MATHEMATICAL PRELIMINARIES AND PROBLEM FORMULATION

  • The actuator-augmented formulation separates actuator state from the physical input and treats actuator-realization defect as a distinct term in the actuator dynamics.The stated condition for input coherence is sufficient but not necessary, and can be met with a second-order low-pass filter when needed.Source: §II · p. 3

III. GALERKIN PROJECTION-BASED FINITE-DIMENSIONAL REALIZATION

  • The finite-dimensional realization uses Gram and cross-Gram matrices for projection, with a positive-definite Gram matrix equivalent to linear independence of the selected functions.The projection formula follows componentwise from Gram normal equations.Source: §III · p. 3

IV. CONTROL-COHERENT FINITE-DIMENSIONAL GALERKIN LIFTING WITH ADDITIVE UNCERTAINTY

The section develops an exact finite-dimensional LTI-plus-residual Galerkin decomposition that preserves the actuator block and constant input matrix at every retained order. It then relates residual bounds to approximation errors and describes conditions for bounding the residual at finite order.

IV. CONTROL-COHERENT FINITE-DIMENSIONAL GALERKIN LIFTING WITH ADDITIVE UNCERTAINTY

  • At every retained order, the Galerkin construction preserves the actuator block and constant input matrix while representing the lifted dynamics as an exact LTI part plus a residual.The construction introduces an actuator-composed lift family and a regularity condition for this decomposition.Source: §IV · p. 4
  • The raw lifts can be reduced to a nested, orthonormal family linearly independent of the actuator-coordinate subspace, with each selected lift a finite real combination of raw lifts.The ordering can be chosen to retain physically useful observables at low orders.Source: §IV · pp. 4–5
  • Along trajectories, the lifted coordinates have a nominal derivative plus an actuator-realization defect, while the lifted state equation retains the exact input term.The nominal projected part is A_Nζ_N, and the exact input term and uncertainty terms yield the lifted state equation.Source: §IV · p. 6
  • The closure and residual depend on the chosen region and measure, whereas the exact actuator blocks do not depend on that weighting.The bounds are therefore regional and measure weighted.Source: §IV · p. 6
  • The closure residual is bounded uniformly over the operating region and its component norms are expressed through best-approximation distances to the reduced space.The stated maxima are finite and attained for the fixed dictionary, coordinates, region, and measure.Source: §IV · p. 6
  • Density for each fixed residual component alone does not control the growing residual vector, while the closed-loop results require only finite-order residual bounds.Uniform decay has a sufficient condition in the appendix, but the closed-loop results do not require that asymptotic limit.Source: §IV · p. 6
  • When exact linear constraints on the residual are available, they can be used to construct structured ellipsoidal enclosures.The residual lies in the nullspace of the constraint matrix and admits an ellipsoidal bound under the stated quadratic condition.Source: §IV · p. 7

A. Generalized plant and controller synthesis

The section embeds the nonlinear lifted dynamics in an exact finite-dimensional closed-loop realization with bounded disturbance channels. It then uses a weighted generalized plant and standard H∞ output-feedback synthesis to obtain a controller for the measured tracking error.

A. Generalized plant and controller synthesis

  • The design formulates a MIMO H∞ output-feedback problem using control input ua and measured tracking signal r − y.The generalized plant interconnects the lifted realization with weighting filters on tracking error, control input, and physical output.Source: A. Generalized plant and controller… · pp. 7–8
  • Before first exit from the operating region, the nonlinear trajectory and its controller and filter states satisfy the closed-loop realization with disturbance channels.The channels represent actuator defect, lower-block uncertainty, and output-readout residual; the actual channels are coupled, so treating them as independent unit-ball inputs can be conservative.Source: A. Generalized plant and controller… · pp. 7–8
  • For a fixed generalized plant, standard H∞ synthesis yields a finite-dimensional proper internally stabilizing controller when the stated conditions hold.The section invokes stabilizability, detectability, feedthrough and rank conditions, and stabilizing Riccati solutions satisfying the coupling condition.Source: A. Generalized plant and controller… · p. 8
  • The closure-channel enclosure can be shaped without changing the nominal feedback interconnection when its additional contraction condition holds.Under that condition, the scalar and shaped closure channels have the same internally stabilizing controller class, and the contraction preserves the stated quadratic bounded-real dissipation inequality.Source: A. Generalized plant and controller… · p. 8

B. Closed-loop analysis

The closed-loop analysis constructs a quadratic storage estimate for a fixed internally stabilizing controller. A strict H∞ bound supplies the Riccati certificate used to derive that estimate.

B. Closed-loop analysis

  • The analysis applies the storage function to the fixed closed-loop realization and its transfer matrix, with a fixed similarity transformation allowed.The closed-loop transfer is identified as the map from exogenous inputs to performance outputs.Source: B. Closed-loop analysis · pp. 8–9
  • For any γN,a greater than the closed-loop H∞ norm, the bounded-real result supplies a positive quadratic storage matrix for the fixed realization.The proof uses a post-synthesis Riccati equation and completes the square to obtain the dissipation estimate.Source: B. Closed-loop analysis · p. 9

C. Original-state bounds

The section converts the closed-loop storage estimate into explicit physical-state and tracking-error bounds before the first exit from the operating region. Sufficient containment checks then extend those estimates to all time and establish forward completeness.

C. Original-state bounds

  • Before first exit, the storage estimate yields explicit transient bounds on the retained physical state and tracking error.The readout gains connect the storage bound to these two quantities.Source: C. Original-state bounds · pp. 9–10
  • Three sufficient localization checks keep the physical state inside the operating region: a performance-storage ellipsoid, an auxiliary ellipsoid, or coordinate-wise bounds.For the auxiliary alternative, the physical-state image of the localized ellipsoid must lie inside the region; for the coordinate-wise alternative, each bound must be strictly below its limit.Source: C. Original-state bounds · p. 10
  • If an applicable containment condition holds, the nonlinear closed loop is forward complete and the pre-exit estimates hold for all time.Compact containment, bounded full closed-loop state, and bounded inputs permit continuation of the locally Lipschitz state equations.Source: C. Original-state bounds · p. 10
  • Under the theorem's hypotheses, the complete closed-loop state is uniformly ultimately bounded.The supplied bound is expressed as a limit superior in terms of γN,a and the uniform outer-channel bound.Source: C. Original-state bounds · pp. 10–11

VI. NUMERICAL EXAMPLE

The numerical example uses an actuator-driven cart–pendulum to illustrate regional certificates and compare output-feedback control with full-state backstepping.

VI. NUMERICAL EXAMPLE

  • The example studies an actuator-driven cart–pendulum with pendulum angle as the measured output.The model specifies angle, angular velocity, and horizontal cart force, and uses the mechanical model cited in the passage.Source: §VI · p. 11
  • The comparison uses the same known actuator-augmented plant for the proposed controller and full-state backstepping.The nonzero closure residual allows regional enclosure and nonlinear guarantees to be considered alongside observed tracking performance.Source: §VI · p. 11

A. Realization, residual, and dictionary degree

The Galerkin realization retains the actuator and input structure while representing finite-order nonclosure as an additive residual. The degree study shows substantial improvement at cubic degree, with smaller gains at degree five because the chosen observables do not capture actuator–plant coupling.

A. Realization, residual, and dictionary degree

  • The cubic-model study uses a fixed region, measure, command, and physical initialization to compare the nonlinear response with the model and residual correction.The Galerkin measure is Lebesgue measure on Ω.Source: A. Realization, residual, and… · p. 11
  • The d = 3 realization reduces maximum angle error from 0.136 to 0.0522 rad and velocity error from 0.0688 to 0.0207 rad/s; d = 5 gives 0.0512 rad and 0.0205 rad/s.The d = 3 realization is used for controller synthesis.Source: A. Realization, residual, and… · p. 11
  • Retaining the actual residual recovers both coordinates to within 3 × 10−14 in numerical integration, while the peak normalized residual is 0.0952.Across 289 paths, 268 remain at 10 s and the largest observed normalized residual is 0.1878.Source: A. Realization, residual, and… · p. 11
  • The chosen dictionary separates actuator coordinates and state polynomials, so increasing state-polynomial degree alone does not capture the input-term coupling b(x1)p1.Products of actuator and plant-state observables are identified as a natural enrichment direction.Source: A. Realization, residual, and… · p. 11

B. Output feedback and regional guarantees

The output-feedback design uses the measured tracking error and a cubic lift to establish regional closed-loop certificates. Under the stated containment condition, the reported simulations satisfy the weighted-performance bound and show bounded tracking errors.

B. Output feedback and regional guarantees

  • The controller is a 14-state design driven only by eN = r − x1, and its stable generalized-plant realization satisfies Assumption 5.3.The synthesis uses two-Riccati design followed by scalar output-gain tuning.Source: B. Output feedback and regional… · p. 11
  • The chain-rule condition (66) is checked along the irregular-reference nonlinear trajectory with x(0) = (0, 0.002), using all 28 closed-loop states.The localization analysis uses a 22-state auxiliary realization based on the same cubic lift and controller.Source: B. Output feedback and regional… · p. 12, Figure 2 · p. 12
  • The stated convolution and homogeneous-response bounds imply all-time containment in Ωloc under the specified reference and initial-state conditions.The reported constants are d1 < 32, d2 < 2450, d3 < 0.70, and d4 < 0.46.Source: B. Output feedback and regional… · p. 12
  • The weighted-output energy satisfies g(65) = 5.573 < 12.1, consistent with the verified bound ∥Tzw,N∥∞ < 12.1.The two energies are integrated with the nonlinear equations.Source: B. Output feedback and regional… · p. 12
  • The irregular-reference simulation reports a maximum angle of 0.2608 rad and tracking error of 3.67 × 10−5 rad over 175 s.Figures 3(b) and 4 show the response and localization.Source: B. Output feedback and regional… · p. 13

C. Backstepping comparison and plant-model uncertainty

The comparison uses full-state backstepping as a nonlinear benchmark against the tracking-error-driven H∞ controller. Their nominal RMSEs are nearly identical, while the reported peak tracking error is lower for output feedback; under both parameter perturbations, its RMSE is also lower.

C. Backstepping comparison and plant-model uncertainty

  • Backstepping cancels nominal nonlinearities using full-state feedback, whereas the proposed controller uses only r − x1; their nominal small-signal tracking bandwidths are made similar.Both controllers use the same 175 s reference and begin with all dynamic states zero.Source: C. Backstepping comparison and… · p. 13
  • The comparison evaluates the time-weighted RMSE over 175 s using trapezoidal quadrature on a common 1 ms output grid.Reference derivatives for backstepping are implemented through the stated filter Qc(s) = 500s/(s + 500).Source: C. Backstepping comparison and… · p. 13
  • Nominal RMSE is 3.23 × 10−5 rad for H∞ feedback and 3.06 × 10−5 rad for backstepping; maximum errors are 6.99 × 10−5 and 8.83 × 10−5 rad, respectively.The paper characterizes the two nominal RMSE values as almost identical.Source: C. Backstepping comparison and… · p. 13
  • In P1, RMSE is 4.69 × 10−5 rad for H∞ feedback and 0.1806 rad for backstepping; in P2, it is 9.75 × 10−5 and 0.2736 rad, respectively.P1 doubles pendulum mass; P2 changes plant parameters and uses a uniform-rod pendulum.Source: C. Backstepping comparison and… · p. 13

VII. CONCLUSION

The paper combines an actuator-preserving Galerkin construction with regional residual bounds for H∞ output-feedback synthesis. The cart–pendulum example reports nominal tracking close to full-state backstepping, with lower peak error, under stated modeling and implementation limits.

VII. CONCLUSION

  • The Galerkin construction preserves the actuator block and constant input matrix while representing finite-order nonclosure as an additive residual.Regional residual enclosures provide uncertainty channels for H∞ output-feedback synthesis.Source: §VII · p. 13
  • Quadratic estimates and exact readout provide physical-state and tracking bounds, while full-state or auxiliary localization yields forward completeness for the certified initial class.Source: §VII · p. 13
  • For the nominal cart–pendulum plant, output feedback achieves almost the same tracking RMSE as full-state backstepping and a lower peak tracking error.Source: §VII · p. 13
  • The analysis assumes a known nonlinear vector field, excludes actuator saturation, and treats residual channels as independent bounded inputs, contributing to conservative bounds.The paper identifies actuator saturation as a principal limitation of the nonlinear extension.Source: §VII · p. 13

APPENDIX A A SUFFICIENT CONDITION FOR GROWING-RESIDUAL

The appendix combines a convergence argument for (90) with closed-loop validation displays and proof steps for Proposition 4.11.

APPENDIX A A SUFFICIENT CONDITION FOR GROWING-RESIDUAL

  • Nesting bounds d_k,N by the distance to W_k, and d_k,N tends to zero for each fixed k; dominated convergence and (46) prove (90).Source: Appendix A · p. 13
  • Fig. 3(a) shows finite-horizon gain for a zero-initial sweep; panel (b) plots normalized plant coordinates against class-wide bounds and the Ωloc boundary.Panel (b) uses the irregular reference and x(0) = (0, 0.002); the bounds come from the cubic auxiliary realization.Source: Figure 3 · p. 14
  • Fig. 4 traces the plant state over 175 s from x(0) = (0, 0.002), with the dashed box marking the plant-state projection of Ωloc.Markers identify 0, 90, and 175 s, and arrowheads show motion direction.Source: Figure 4 · p. 14
  • Fig. 5 compares nominal and perturbed-plant responses over 175 s while both controllers retain their nominal parameters in P1 and P2.The inset enlarges nominal responses and the reference over 34.70–34.80 s.Source: Figure 5 · p. 14

APPENDIX B PROOF OF PROPOSITION 4.11

  • The Proposition 4.11 proof establishes that eΠ_WN is a bounded projection with range WN, then applies a standard projection estimate.The estimate bounds the error by (1 + ΛN) times the best uniform approximation error in WN.Source: Appendix B · p. 14
  • The proof obtains (51) by bounding residuals pointwise over Ω, taking the maximum, and then taking square roots.Source: Appendix B · p. 14

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