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Online Simultaneous State and Parameter Estimation for Second-order Nonlinear Systems

Rushikesh Kamalapurkar

TL;DR

The paper tackles online simultaneous state and parameter estimation for a class of second-order nonlinear systems with uncertain dynamics, using an output-feedback concurrent learning approach that relaxes the persistent excitation requirement. It couples an adaptive velocity observer with a data-driven parameter estimator based on a stored history stack and a dwell-time purging mechanism, underpinned by a Lyapunov-based stability analysis. The key contributions include an integral-error affine formulation $P(t)=F(t)+\theta^{T}G(t)+E(t)$, a derivative-free velocity estimator, a least-squares–type parameter update with data management, and a switched-system stability framework guaranteeing uniform ultimate boundedness of the estimation errors. Simulation on a two-link manipulator demonstrates robustness to measurement noise and effective convergence of state and parameter estimates, highlighting practical applicability to robotic and vehicle systems.

Abstract

In this paper, a concurrent learning based adaptive observer is developed for a class of second-order nonlinear time-invariant systems with uncertain dynamics. The developed technique results in simultaneous online state and parameter estimation. A Lyapunov-based analysis is used to show that the state and parameter estimation errors are uniformly ultimately bounded. As opposed to persistent excitation which is required for parameter estimation in traditional adaptive control methods, the developed technique only requires excitation over a finite time interval.

Online Simultaneous State and Parameter Estimation for Second-order Nonlinear Systems

TL;DR

The paper tackles online simultaneous state and parameter estimation for a class of second-order nonlinear systems with uncertain dynamics, using an output-feedback concurrent learning approach that relaxes the persistent excitation requirement. It couples an adaptive velocity observer with a data-driven parameter estimator based on a stored history stack and a dwell-time purging mechanism, underpinned by a Lyapunov-based stability analysis. The key contributions include an integral-error affine formulation , a derivative-free velocity estimator, a least-squares–type parameter update with data management, and a switched-system stability framework guaranteeing uniform ultimate boundedness of the estimation errors. Simulation on a two-link manipulator demonstrates robustness to measurement noise and effective convergence of state and parameter estimates, highlighting practical applicability to robotic and vehicle systems.

Abstract

In this paper, a concurrent learning based adaptive observer is developed for a class of second-order nonlinear time-invariant systems with uncertain dynamics. The developed technique results in simultaneous online state and parameter estimation. A Lyapunov-based analysis is used to show that the state and parameter estimation errors are uniformly ultimately bounded. As opposed to persistent excitation which is required for parameter estimation in traditional adaptive control methods, the developed technique only requires excitation over a finite time interval.

Paper Structure

This paper contains 8 sections, 1 theorem, 48 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Error System for Estimation
  3. Velocity estimator design
  4. Parameter Estimator Design
  5. Purging
  6. Stability Analysis
  7. Simulation
  8. Conclusion

Key Result

Theorem 1

Let $\varepsilon>0$ be given. Let the history stacks $\mathcal{H}$ and $\mathcal{G}$ be populated using the algorithm detailed in Fig. alg:CLNoXDotpurgeDwell. Let the learning gains be selected to satisfy the sufficient gain conditions in (eq:V Gain Conditions 1), (eq:V gain condition), (eq:W Gain C $\mathcal{T}_{s}$ is selected to be large enough to satisfy (eq:Dwell). Furthermore assume that the

Figures (7)

  • Figure 1: Algorithm for history stack purging with dwell time. At each time instance $t$, $\delta\left(t\right)$ stores the last time instance $\mathcal{H}$ was purged, $\eta\left(t\right)$ stores the highest minimum singular value of $\mathscr{G}$ encountered so far, $\mathcal{T}\left(t\right)$ denotes the dwell time, and $\xi\in\left(0,1\right]$ denotes a threshold fraction.
  • Figure 2: Trajectories of the parameter estimation errors using noise-free position measurements.
  • Figure 3: Trajectories of the generalized position estimation errors using noise-free position measurements.
  • Figure 4: Trajectories of the generalized velocity estimation errors using noise-free position measurements.
  • Figure 5: Trajectories of the parameter estimation errors with a Gaussian measurement noise (variance = 0.001).
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • proof