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Beyond Bounded Noise: Stochastic Set-Membership Estimation for Nonlinear Systems

Felix Brändle, Nicolas Chatzikiriakos, Andrea Iannelli, Frank Allgöwer

Abstract

In this paper, we derive a novel procedure for set-membership estimation of dynamical systems affected by stochastic noise with unbounded support. By employing a bound on the sample covariance matrix, we are able to provide a finite-sample uncertainty set containing the true system parameters with high probability. Our approach can be natively applied to a wide class of nonlinear systems affected by sub- Gaussian noise. Through our analysis, we provide conditions under which the proposed uncertainty set converges to the true system parameters and establish an upper bound on the convergence rate. The proposed uncertainty set can be used directly for the synthesis of robust controllers with probabilistic stability and performance guarantees. Concluding numerical examples demonstrate the advantages of the proposed formulation over established approaches.

Beyond Bounded Noise: Stochastic Set-Membership Estimation for Nonlinear Systems

Abstract

In this paper, we derive a novel procedure for set-membership estimation of dynamical systems affected by stochastic noise with unbounded support. By employing a bound on the sample covariance matrix, we are able to provide a finite-sample uncertainty set containing the true system parameters with high probability. Our approach can be natively applied to a wide class of nonlinear systems affected by sub- Gaussian noise. Through our analysis, we provide conditions under which the proposed uncertainty set converges to the true system parameters and establish an upper bound on the convergence rate. The proposed uncertainty set can be used directly for the synthesis of robust controllers with probabilistic stability and performance guarantees. Concluding numerical examples demonstrate the advantages of the proposed formulation over established approaches.

Paper Structure

This paper contains 8 sections, 5 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Set-Membership estimation
  4. High-probability set-membership estimation
  5. Asymptotic convergence
  6. Convergence speed
  7. Numerical evaluation
  8. Conclusion

Key Result

Theorem 1

Let $W^\top\in\mathbb{R}^{N\times n_x}$ be a matrix whose rows $w_i^\top$ are independent sub-gaussian isotropic random vectors with zero mean in $\mathbb{R}^{n_x}$. Then for every $N\geq 1$, with probability at least $1-\delta$ it holds that where $c_1>0$ and $c_2>0$ are constants that depend only on the distribution of $w_i$ and $W=[w_1,\ldots,w_N]$. Furthermore, with the same probability, it h

Figures (2)

  • Figure 3: Mean volume of uncertainty sets for the LTI system.
  • Figure 4: Mean volume of uncertainty sets for the nonlinear system.

Theorems & Definitions (11)

  • Theorem 1: Theorem 39, vershynin2012
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Lemma 4
  • ...and 1 more