Identifiability and Observability of Nonsmooth Systems via Taylor-like Approximations

Abstract

New sensitivity-based methods are developed for determining identifiability and observability of nonsmooth input-output systems. More specifically, lexicographic calculus is used to construct nonsmooth sensitivity rank condition (SERC) tests, which we call lexicographic SERC (L-SERC) tests. The introduced L-SERC tests are: (i) practically implementable and amenable to large-scale problems; (ii) accurate since they directly treat the nonsmoothness while avoiding, e.g., smoothing approximations; and (iii) analogous to (and indeed recover) their smooth counterparts. To accomplish this, a first-order Taylor-like approximation theory is developed using lexicographic differentiation to directly treat nonsmooth functions. A practically implementable algorithm is proposed that determines partial structural identifiability or observability, a useful characterization in the nonsmooth setting. Lastly, the theory is illustrated through an application in climate modeling.

Figures (2)

• Figure 1: Trajectories of the state variables $T$, $V$, and the output $y$. The output $y$ crosses a nonsmooth threshold when $T(t)=T_{\min}$ (indicated by the first vertical dashed line). The second vertical dashed line corresponds to a nonsmooth threshold in the model being crossed when $T(t)=V(t)$.
• Figure 2: This figure presents the output sensitivity function $\caps_{\y}^{\rm L}(t)=\mathrm{lshift}(\capy^*(t)) \in \real^{1 \times 3}$, with $\capy^*(t)$ from \ref{['eq:Stommel_X_dot']} for identifiability (left panel), and $\caps_{\y}^{\rm L}(t)=\mathrm{lshift}(\capy^*(t)) \in \real^{1 \times 2}$, with $\capy^*(t)$ from \ref{['eq:Stommel_X_dot_obs']} for observability (right panel). When computing $\caps_{\y}(t)$ for both the identifiability and observability, we can see that, as shown in Figure \ref{['fig:Stommel Solution vs Output']}, the nonsmoothness happens in the output when $T(t)=T_{\min}$ (first dashed line) and in the model when $T(t)=V(t)$ (second dashed line). The stars in this figure represent the sample times $\{t_k\}$ used to construct ${\pmb \Upsilon}_{\dd}$ in \ref{['LSERC_ID']}.

Theorems & Definitions (13)

• theorem 1
• corollary 1
• remark 1
• definition 1
• definition 2
• theorem 2
• definition 3
• definition 4
• theorem 3
• remark 2