Identification of Analytic Nonlinear Dynamical Systems with Non-asymptotic Guarantees

Abstract

This paper focuses on the system identification of an important class of nonlinear systems: linearly parameterized nonlinear systems, which enjoys wide applications in robotics and other mechanical systems. We consider two system identification methods: least-squares estimation (LSE), which is a point estimation method; and set-membership estimation (SME), which estimates an uncertainty set that contains the true parameters. We provide non-asymptotic convergence rates for LSE and SME under i.i.d. control inputs and control policies with i.i.d. random perturbations, both of which are considered as non-active-exploration inputs. Compared with the counter-example based on piecewise-affine systems in the literature, the success of non-active exploration in our setting relies on a key assumption on the system dynamics: we require the system functions to be real-analytic. Our results, together with the piecewise-affine counter-example, reveal the importance of differentiability in nonlinear system identification through non-active exploration. Lastly, we numerically compare our theoretical bounds with the empirical performance of LSE and SME on a pendulum example and a quadrotor example.

Figures (4)

• Figure 1: Convergence rate of the LSE for pendulum and quadrotor scenarios: (a) Pendulum example with uniform, (b) Pendulum example with truncated-Gaussian, (c) Quadrotor example with uniform, and (d) Quadrotor example with truncated-Gaussian noises and disturbances. Here, uniform noises and disturbances are i.i.d. generated from $\texttt{uniform}([-1, 1])$, and truncated-Gaussian noises and disturbances are i.i.d. generated from $\texttt{truncated-Gaussian}(0, 0.1, [-1, 1])$. "theo" denotes the theoretical convergence rate, and "empr" represents the empirical rate. The mean error across 20 trials is shown by dots on the empirical plots, with shaded areas illustrating empirical standard deviation.
• Figure 2: Convergence rate of the SME for pendulum and quadrotor scenarios: (a) Pendulum example with uniform, (b) Pendulum example with truncated-Gaussian, (c) Quadrotor example with uniform, and (d) Quadrotor example with truncated-Gaussian noises and disturbances. Here, uniform noises and disturbances are i.i.d. generated from $\texttt{uniform}([-1, 1])$, and truncated-Gaussian noises and disturbances are i.i.d. generated from $\texttt{truncated-Gaussian}(0, 0.5, [-1, 1])$. "theo" denotes the theoretical convergence rate, and "empr" represents the empirical rate. The mean error across 10 trials is shown by dots on the empirical plots, with shaded areas illustrating empirical standard deviation.
• Figure 3: Performance of SME for pendulum in (a) with control input $u_{t} = -k\dot{\alpha}_{t}+\eta_{t}$ where $k=0.1$, $\eta_{t}$ i.i.d. generated from $\texttt{truncated-Gaussian}(0, 2, [-2, 2])$ and disturbed with $w_{t}$ i.i.d. generated from $\texttt{truncated-Gaussian}(0, 1, [-1, 1])$. (b) Diameter of the uncertainty set estimated by SME. (c) Uncertainty set depicted for $T=50, 200, 250, 400, 500$.
• Figure 4: 2D projections of the uncertainty set estimated by SME for the unknown parameters of the quadrotor example. The noises and disturbances are i.i.d generated from $\texttt{truncated-Gaussian}(0,0.5,[-1,1])$.

Theorems & Definitions (23)

• Example 1: Pendulum
• Example 2: Quadrotor alaimo2013mathematical
• Definition 1: Semi-continuous distribution
• Remark 1: Connection with Lebesgue Decomposition Theorem
• Definition 2: Locally input-to-state stability (LISS)
• Definition 3: Persistent excitation skantze2000adaptivesastry2011adaptive
• Definition 4: BMSB simchowitz2018learning
• Theorem 1: BMSB for open-loop systems
• Theorem 2: LSE's convergence rate for open-loop systems
• Corollary 1: LSE's convergence rate for closed-loop systems