Robust Hybrid Finite-Time Parameter Estimation Without Persistence of Excitation

1. Problem Statement

The paper addresses the problem of estimating unknown parameters ($\theta^*$) in a linear regression model of the form:
$$y(t) = \theta^{*\top}\phi(t)$$
where $y(t)$ is the output and $\phi(t)$ is the known regressor.

Key Challenges:

• Convergence Speed: Classical gradient descent algorithms guarantee exponential convergence only under the Persistency of Excitation (PE) condition, which requires the regressor to be exciting over every interval of a fixed length.
• Relaxed Excitation: In many practical scenarios, the regressor is only exciting over a finite time interval (e.g., a specific window of data) rather than persistently. Under these conditions, classical methods fail to converge to the exact parameter value.
• Noise Robustness: Estimators must remain stable and provide bounded errors in the presence of measurement noise.
• Computational Scalability: Existing finite-time methods (like DREM-based approaches) often suffer from high computational costs ($O(n^3)$) at every time step due to matrix inversions, making them unsuitable for high-dimensional systems.

2. Methodology

The authors propose a Hybrid Systems Framework that combines continuous-time flow dynamics with discrete-time jumps to achieve finite-time convergence.

A. Hybrid Estimator Architecture

The estimator utilizes two coupled parameter estimates, $\theta_1$ and $\theta_2$, evolving continuously with different adaptation rates ($\gamma_1 \neq \gamma_2$).

• Flow Dynamics (Continuous): Between jumps, the estimates evolve via gradient descent:
  $$\dot{\theta}_i = -\gamma_i \phi(t)(\phi(t)^\top \theta_i - y(t))$$
• Jump Dynamics (Discrete): At specific time intervals (triggered by a timer $\tau_a = \delta$), a discrete jump occurs. The estimates are updated using a linear combination of the pre-jump values:
  $$\theta_i^+ = K_1(q)\theta_1 + K_2(q)\theta_2$$
  where the gains $K_1$ and $K_2$ are calculated based on the state-transition matrices ($\Phi_1, \Phi_2$) of the two continuous flows over the interval $[0, \delta]$.

B. Excitation Conditions

The core theoretical innovation is the relaxation of the excitation requirement:

• Constant Parameters: The regressor $\phi(t)$ only needs to be exciting over a single finite interval $[0, \delta]$. The algorithm guarantees exact convergence at the first jump time $t=\delta$.
• Piecewise Constant Parameters: For parameters that change values over time, the classical PE condition is required, but the algorithm ensures convergence within a bounded time (specifically, within $2\delta$ after a parameter change).

C. Robustness Analysis

The paper analyzes robustness against measurement noise $w(t)$ using Input-to-State Stability (ISS) and Integral Input-to-State Stability (iISS):

• Under PE conditions, the estimator is shown to be ISS with respect to bounded noise.
• Under Finite-Time Excitation (relaxed condition), the estimator is shown to be iISS, meaning the estimation error depends on the integral of the noise magnitude rather than its peak value.

D. Computational Efficiency

Unlike Dynamic Regressor Extension and Mixing (DREM) methods that require computing determinants and matrix inverses at every time step ($O(n^3)$ cost), the proposed hybrid estimator:

• Performs simple gradient descent ($O(n)$) during continuous flows.
• Only performs the expensive $O(n^3)$ matrix inversion and state-transition computation once per interval $\delta$ (at the jump instants).

3. Key Contributions

1. Finite-Time Convergence without PE: The first hybrid estimator to guarantee exact finite-time convergence for constant parameters using only a finite-time excitation condition, eliminating the need for persistent excitation.
2. Two-Estimator Coupling: Introduction of a dual-estimator structure with distinct adaptation rates to construct a jump map that cancels out estimation errors exactly at the jump instant.
3. Robustness Guarantees: Formal proof of ISS and iISS properties, demonstrating that the estimator remains stable and bounded under measurement noise.
4. Scalability: A significant reduction in computational complexity for high-dimensional systems by decoupling the $O(n^3)$ operations from the continuous time step.
5. Extension to Time-Varying Parameters: A method to handle time-varying parameters by decomposing them into basis functions, transforming the problem into one with constant unknown coefficients.

4. Results

• Theoretical Proofs: The paper provides rigorous proofs (using Lyapunov functions and hybrid systems theory) showing that the parameter error converges to zero exactly at $t=\delta$ for constant parameters and within $2\delta$ for piecewise constant parameters.
• Numerical Simulations (Constant Parameters):
  • Demonstrated convergence to exact values in $\delta=1$ second for a system where the regressor was only exciting on $[0, 2]$ and zero elsewhere.
  • Classical gradient descent failed to converge in this scenario.
  • In the presence of noise ($w(t) = 6\sin(10t)$), the estimation error remained bounded proportional to the noise magnitude.
• Numerical Simulations (Time-Varying Parameters): Successfully estimated a time-varying parameter generated by polynomial basis functions, reconstructing the exact trajectory after the excitation interval.
• Scalability Benchmarks:
  • Compared against DREM-based approaches (Refs [14], [5], [17]) across dimensions $n=10$ to $n=5000$.
  • Speedup: The hybrid approach showed a speedup ratio increasing from 1.24x at $n=10$ to 906x at $n=5000$.
  • At $n=5000$, the DREM approaches took over 1 hour, while the hybrid estimator took only 4.94 seconds.

5. Significance

This work represents a significant advancement in adaptive control and system identification:

• Practical Applicability: It enables accurate parameter estimation in scenarios where data is scarce or excitation is transient (e.g., short-duration maneuvers, specific sensor windows), which was previously impossible with finite-time guarantees.
• High-Dimensional Viability: By drastically reducing computational overhead, it makes finite-time parameter estimation feasible for large-scale systems (e.g., robotics, power grids, high-dimensional signal processing) where previous methods were computationally prohibitive.
• Theoretical Rigor: It bridges the gap between hybrid systems theory and adaptive control, providing formal robustness guarantees (ISS/iISS) that are often lacking in heuristic finite-time estimators.

In summary, the proposed algorithm offers a computationally efficient, robust, and theoretically sound solution for parameter estimation that relaxes the strict persistence of excitation requirements while guaranteeing exact convergence in predetermined finite time.