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New Results on Parameter Estimation via Dynamic Regressor Extension and Mixing: Continuous and Discrete-time Cases
Romeo Ortega, Stanislav Aranovskiy, Anton A. Pyrkin, Alessandro Astolfi, Alexey A. Bobtsov
TL;DR
The paper addresses parameter estimation for linear regression models, where classical methods rely on restrictive excitation conditions. It develops unified CT and DT DREM results, new regressor extensions, and a finite-time estimator. The results include weaker-than-PE convergence conditions, quantifiable transient improvement, and finite-time estimation that remains alert to time-varying parameters.
Problem
Classical gradient and least-squares parameter estimators rely on the restrictive persistent-excitation condition, motivating estimators that relax this requirement.
Method
The paper develops DREM estimators for CT and DT linear regression models, proposes new extended regressor operators, and introduces an alternative finite-time estimator.
Results
The proposed results provide convergence under excitation conditions strictly weaker than regressor PE, quantifiable transient performance improvement, and finite-time estimation that retains alertness to track time-varying parameters.
Takeaways & Limitations
DREM offers a unified parameter-estimation framework with componentwise monotonicity, convergence without PE under alternative conditions, and finite-time estimation capabilities.
Abstract
from arXiv · showhide
We present some new results on the dynamic regressor extension and mixing parameter estimators for linear regression models recently proposed in the literature. This technique has proven instrumental in the solution of several open problems in system identification and adaptive control. The new results include: (i) a unified treatment of the continuous and the discrete-time cases; (ii) the proposal of two new extended regressor matrices, one which guarantees a quantifiable transient performance improvement, and the other exponential convergence under conditions that are strictly weaker than regressor persistence of excitation; and (iii) an alternative estimator ensuring parameter estimation in finite-time that retains its alertness to track time-varying parameters. Simulations that illustrate our results are also presented.
I. INTRODUCTION
The paper studies parameter estimation for linear regression models, where classical gradient and least-squares methods depend on restrictive persistent excitation. It introduces DREM as a framework that relaxes this requirement and develops unified continuous- and discrete-time results.
- Classical gradient and least-squares estimators require persistent excitation for parameter convergence in linear regression models.
- DREM extends the regressor and mixes the resulting equations to address limitations of classical estimators in continuous- and discrete-time systems.
- DREM guarantees monotonicity of each parameter error component, rather than only monotonicity of the parameter-error norm.
- DREM establishes parameter convergence without PE by imposing a non-square-integrability condition on a designer-dependent extended regressor determinant.
- DREM can generate finite-time-convergent estimates under interval excitation, while the paper develops results for both CT and DT cases.
B. Generation of m scalar LRE via DREM
DREM constructs an extended matrix regression and then uses adjugate-based mixing to produce independent scalar regressions for the individual parameters. Earlier extensions lacked a quantitative transient-performance advantage over gradient estimation.
- A bounded-input bounded-output stable linear operator creates the extended output vector and regressor matrix used by DREM.
- Adjugate multiplication converts the matrix regression into m scalar equations, each relating one transformed output to one parameter.
- Earlier LTI-filter extensions had simulation evidence but no established quantitative advantage over gradient estimation.
C. Properties of gradient parameter estimators in DREM
The scalar regressions generated by DREM enable stronger gradient-estimator properties than standard vector-based estimation. These include componentwise monotonicity, convergence under weaker excitation conditions, and adjustable convergence rates.
- The scalar LREs are the central DREM feature enabling stronger parameter-estimation results with simple gradient estimators.
- If Δ belongs to PE, the DREM-based scalar estimators converge exponentially.
- Each individual parameter error is monotonically non-increasing, which is stronger than monotonicity of the parameter-error norm.
- DREM replaces the restrictive PE requirement with a non-square-integrability or summability condition for parameter convergence.
- DREM convergence rates can be made arbitrarily fast by increasing γ_i in CT or decreasing it in DT.
III. A DREM ESTIMATOR WITH STRICTLY WEAKER CONVERGENCE CONDITIONS
This section presents a DREM construction whose parameter-convergence conditions are strictly weaker than regressor persistence of excitation, including exponential convergence under a precise implication structure.
- The proposed DREM version establishes convergence under excitation conditions strictly weaker than φ ∈PE.
- The discrete-time construction uses an LTV operator to generate the extended regressor and scalar gradient-descent estimators.
- Parameter convergence requires ∆(k) ∉ℓ2, a condition stated to be strictly weaker than φ(k) ∈PE.
- Exponential parameter convergence requires ∆(k) ∈PE, which is also weaker than φ(k) ∈PE under the stated window-size qualifiers.
- The proof relies on implications relating persistence of excitation of φ(k), ∆(k), and the extended-regressor window sizes.
- The qualifiers K ≥2 and K ≤K̄ are necessary; without them, ∆(k) ∈PE implies φ(k) ∈PE.
IV. SOME SPECIFIC CHOICES OF THE OPERATOR H
This section generalizes the DREM operator using bounded, stable LTV systems, recovers Kreisselmeier’s regressor extension as a special case, and relates the two constructions explicitly.
- The paper proposes general LTV operators beyond first-order LTI filters and pure delays used in earlier DREM work.
- The operators use time-varying state-space matrices and feedforward terms, with stable realizations and bounded matrices as requirements.
- For bounded realization matrices, BIBO-stable LTV subsystems are also globally exponentially stable.
- Kreisselmeier’s regressor extension filters φy and φφ⊤ with one BIBO-stable operator, whereas DREM filters y and φ with m different operators.
- KRE is a particular case of the generalized DREM construction under the specified operator parameters, yielding Z = Y and Ω = Φ.
- The resulting extended linear regression is used in recently proposed model-reference adaptive controllers.
C. An operator H with guaranteed transient performance improvement
The section analyzes DREM transient behavior through the determinant of the extended regressor and proposes operator choices that produce a quantifiable performance improvement.
- The explicit parameter-error solutions characterize the time evolution of DREM estimation errors in both continuous and discrete time.
- DREM transient performance is determined by the size of ∆^2, with faster convergence obtained for a larger ∆^2.
- The proposed improvement selects feedforward gains d_i in the LTV operators H_i to modify the extended regressor.
- The transient-performance comparison uses parameter-error trajectories from zero-feedforward and proposed operator choices with other parameters and initial conditions held fixed.
- The proof uses operator definitions, Sylvester’s determinant formula, and positivity of the resulting determinant-related term.
V. CT DREM ESTIMATORS WITH ALERT FINITE-TIME CONVERGENCE
The paper recalls that continuous-time DREM can produce finite-time-convergent estimates under the weakest interval excitation assumption.
- Continuous-time DREM can generate finite-time-convergent parameter estimates under the weakest interval excitation assumption.
A. An FTC DREM
This section presents an FTC estimator for scalar continuous-time linear regression equations and establishes finite-time convergence under its stated conditions. It also identifies a trajectory-dependent robustness limitation.
- A. An FTC DREM: The estimator achieves parameter estimation error convergence to zero in finite time.The result is stated for the scalar continuous-time LRE and gradient-descent estimator under the proposition’s assumptions.
- A. An FTC DREM: The FTC property is trajectory-dependent because it concerns only the trajectory generated from the initial condition w_i(0) = 1.Other closed-loop trajectories may exist.
- A. An FTC DREM: Perturbations may drive the favorable trajectory toward an unfavorable one, creating a robustness problem requiring further investigation.
B. New FTC DREM
This section introduces an alternative FTC DREM estimator designed to avoid the standard estimator’s loss of finite-time alertness. Its key mechanism allows excitation increases to restore finite-time tracking after parameter changes, with a gain–settling-time trade-off.
- B. New FTC DREM: The standard FTC approach loses finite-time alertness because its signal w(t) is monotonically non-increasing and generally converges to zero.After this occurs, the estimator reduces to the standard gradient estimator and requires resetting to track parameter variations in finite time.
- B. New FTC DREM: The proposed alternative avoids the practical drawback of resetting the estimator to retain finite-time tracking after new excitation arrives.
- B. New FTC DREM: The new signal w_i^D(t) grows when the determinant increases over an interval of length T_D, indicating arrival of new excitation.This distinguishes it from the always non-increasing signal used by the earlier FTC estimator.
- B. New FTC DREM: The new FTC estimator preserves finite-time convergence when parameters change by responding to newly arriving excitation.
- B. New FTC DREM: Choosing μ_i involves a trade-off between high-gain injection near μ_i = 1 and the time required to achieve finite-time convergence.
VI. SIMULATIONS
The simulations compare gradient and DREM estimators under sufficiently rich and non-PE inputs, then test finite-time tracking with time-varying parameters. DREM is consistent in both input scenarios, the feedforward extension improves transients, and the new FTC estimator retains alertness after parameter changes.
- VI. SIMULATIONS: Both DREM schemes yield consistent estimates with sufficiently rich and constant plant inputs, whereas the gradient scheme has a significant steady-state error for the constant input.The compared inputs are u(t) = 15 sin(2.5t + 1) and u(t) = 15.
- VI. SIMULATIONS: Less than one second is required for parameter convergence with the feedforward term, compared with almost two seconds for DREM with d = 0.The feedforward term is d(t) from (28).
- VI. SIMULATIONS: Both FTC estimators converge in finite time in the initial comparison, while the gradient estimator converges only asymptotically.
- VI. SIMULATIONS: The new FTC estimator preserves finite-time alertness after the first parameter jump and rapidly tracks the linearly time-varying parameter.
- VI. SIMULATIONS: After the parameter change at t = 10, the new FTC tracks the jump in finite time, while neither estimator tracks the ramp under insufficient excitation.The new FTC nevertheless performs much better during the ramp change.
VII. FUTURE WORK
Future work extends the new results toward discrete-time settings, further studies the operator H and additive signals, and addresses nonlinear parameterizations.
- The authors plan to derive results currently available only for continuous time for the practically important discrete-time case.
- They will further explore how the operator H affects the determinant of the extended regressor matrix Φ.
- The authors plan to study additive signals in the linear regression equation through their input-to-state stability properties.
- Handling general nonlinear parameterizations remains a challenging open problem beyond the separable nonlinearities to which DREM is directly applicable.Preliminary results exist for convex, concave, or monotone cases, while the more general case remains open.