# Partial Excitation in Parameter Learning

arXiv: https://arxiv.org/abs/2503.02235 · HTML: https://mlsanity.com/paper/2503.02235/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 69 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Concurrent learning for parameter estimation using dynamic state-derivative estimators](https://mlsanity.com/paper/1507.08903/) (2015)
- [New Results on Parameter Estimation via Dynamic Regressor Extension and Mixing: Continuous and Discrete-time Cases](https://mlsanity.com/paper/1908.05125/) (2019)
- [A nonparametric learning framework for nonlinear robust output regulation](https://mlsanity.com/paper/2309.14645/) (2023)
- [Nonparametric Steady-State Learning for Nonlinear Output Feedback Regulation](https://mlsanity.com/paper/2402.16170/) (2024)
- [Performance Enhancement of Parameter Estimators via Dynamic Regressor Extension and Mixing](https://mlsanity.com/paper/1509.02763/) (2015)
- [Modal Barriers to Controllability in Networks with Linearly-Coupled Homogeneous Subsystems](https://mlsanity.com/paper/1805.01995/) (2018)
- [Identifiability Implies Robust, Globally Exponentially Convergent On-line Parameter Estimation: Application to Model Reference Adaptive Control](https://mlsanity.com/paper/2108.08436/) (2021)
- [Relaxation of Conditions for Convergence of Dynamic Regressor Extension and Mixing Procedure](https://mlsanity.com/paper/2112.04548/) (2021)
- [Distributed State Estimation for Jointly Observable Linear Systems over Time-varying Networks](https://mlsanity.com/paper/2302.12161/) (2023)
- [Stability of FFLS-based diffusion adaptive filter under a cooperative excitation condition](https://mlsanity.com/paper/2310.06287/) (2023)
- [Distributed Parameter Estimation with Gaussian Observation Noises in Time-varying Digraphs](https://mlsanity.com/paper/2311.03911/) (2023)

## Abstract (arXiv)

This paper investigates parameter learning problems under Partial Persistent Excitation (PPE). The PPE condition is a rank-deficient, and therefore, a more general evolution of the well-known Persistent Excitation (PE) condition. Under the PPE condition, a proposed online algorithm is able to calculate the PE and non-PE subspaces, and finally gives an optimal parameter estimate in the sense of least squares. In particular, the learning error within the PE subspace exponentially converges to zero in the noise-free case. The PPE condition also provides a new perspective for solving distributed parameter learning problems, where the challenge is posed by local regressors that are often insufficiently excited. To improve knowledge of the unknown parameters, a cooperative learning protocol is proposed for a group of estimators that collect measured information under complementary PPE condition. This protocol allows each local estimator to operate locally in its PE subspace, and reach a consensus with neighbors in its non-PE subspace. As a result, the task of estimating unknown parameters can be achieved in a distributed way using cooperative local estimators. Application examples in system identification are given to demonstrate the effectiveness of the theoretical results developed in this paper.