On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems

Abstract

In the paper, we consider the problem of robust approximation of transfer Koopman and Perron–Frobenius (P–F) operators from noisy time-series data. In most applications, the time-series data obtained from simulation or experiment are corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite-dimensional approximation of the deterministic system to a random uncertain case. However, these results hold only in asymptotic and under the assumption of infinite data set. In practice, the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min–max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least-square problem. This equivalence between robust optimization problem and regularized least-square problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent trade-offs between the quality of approximation and complexity of approximation. These trade-offs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than some of the existing algorithms like extended dynamic mode decomposition (EDMD), subspace DMD, noise-corrected DMD, and total DMD for systems with process and measurement noise.

Appendix

1.1 A. Proof of Theorem 6

Proof

The min–max optimization problem is

$$\begin{aligned} \mathcal{J} = \min _{\mathbf{K}}\max _{\delta \mathbf{G},\delta \mathbf{A}\in {\bar{\Delta }}}\parallel (\mathbf{G}+\delta \mathbf{G})\mathbf{K}-(\mathbf{A}+\delta \mathbf{A})\parallel _F \end{aligned}$$

(39)

Fix $\mathbf{K}\in \mathbb {R}^{L\times L}$ and let

$$\begin{aligned} r = \max _{\delta \mathbf{G},\delta \mathbf{A}\in {\bar{\Delta }}}\parallel (\mathbf{G}+\delta \mathbf{G})\mathbf{K}-(\mathbf{A}+\delta \mathbf{A})\parallel _F \end{aligned}$$

be the worst-case residual. Then,

$$\begin{aligned} r\le & {} \max _{\delta \mathbf{G},\delta \mathbf{A}\in {\bar{\Delta }}} \parallel \mathbf{G} \mathbf{K} - \mathbf{A} \parallel _F + \parallel \delta \mathbf{G}{} \mathbf{K} - \delta \mathbf{A} \parallel _F\nonumber \\\le & {} \parallel \mathbf{G} \mathbf{K} - \mathbf{A} \parallel _F + \lambda \parallel \mathbf{K} - I\parallel _F\nonumber \\\le & {} \parallel \mathbf{G} \mathbf{K} - \mathbf{A} \parallel _F + \lambda \sqrt{\parallel \mathbf{K}\parallel _F^2 + L} \end{aligned}$$

(40)

Now, for a $L\times L$ matrix $M=[m_{i,j}]\in \mathbb {R}^{L\times L}$, let $\hbox {vec}(M)$ denote the vector

$$\begin{aligned} \hbox {vec}(M)=[m_{1,1},\ldots ,m_{L,1},m_{1,2},\ldots ,m_{L,2},\ldots , m_{L,L}]^\top . \end{aligned}$$

(41)

This follows from the fact that $\mathbb {R}^{L\times L}:= \mathbb {R}^L\otimes \mathbb {R}^L \cong \mathbb {R}^{L^2}$. Hence, $\parallel M\parallel _F = \parallel \hbox {vec}(M)\parallel _2$.

Let

$$\begin{aligned} \begin{pmatrix} \hbox {vec}(\delta \mathbf{G}^\star )&\hbox {vec}(\delta \mathbf{A}_i^\star ) \end{pmatrix} = \frac{\lambda u}{\sqrt{\parallel \mathbf{K}_F^2\parallel _2 + L}}\begin{pmatrix} \hbox {vec}(\mathbf{K})^\top&L \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} u = {\left\{ \begin{array}{ll} \frac{\hbox {vec}(\mathbf{G}{} \mathbf{K}-\mathbf{A})}{\parallel \mathbf{G}{} \mathbf{K}-\mathbf{A} \parallel _F}, if \quad \mathbf{G}{} \mathbf{K}_i\ne \mathbf{A}_i\\ \text {any unit norm vector otherwise.} \end{array}\right. } \end{aligned}$$

Then,

$$\begin{aligned} r\ge & {} \parallel \hbox {vec}(\mathbf{G}{} \mathbf{K} - \mathbf{A}) + \hbox {vec}(\delta \mathbf{G}^\star \mathbf{K} - \delta \mathbf{A}^\star ) \parallel _2\nonumber \\\ge & {} \parallel \hbox {vec}(\mathbf{G}{} \mathbf{K} - \mathbf{A}) \nonumber \\&+ \frac{\lambda \times \hbox {vec}(\mathbf{G}{} \mathbf{K} - \mathbf{A})}{{\sqrt{\parallel \mathbf{K}_F^2\parallel _2 + L}}\parallel \mathbf{G}{} \mathbf{K} - \mathbf{A}\parallel _F}(\parallel \mathbf{K}\parallel _F^2 +L) \parallel _2\nonumber \\\ge & {} \parallel \hbox {vec}(\mathbf{G}{} \mathbf{K} - \mathbf{A}) \parallel _2 + \lambda \sqrt{\parallel \mathbf{K}\parallel _F^2 +L}\nonumber \\= & {} \parallel \mathbf{G}{} \mathbf{K} - \mathbf{A} \parallel _F + \lambda \sqrt{\parallel \mathbf{K}\parallel _F^2 +L}. \end{aligned}$$

(42)

Hence, from (40) and (42), the worst-case residual (24) is

$$\begin{aligned} r = \min _{\mathbf{K}}\parallel \mathbf{G}{} \mathbf{K} - \mathbf{A} \parallel _F + \lambda \sqrt{\parallel \mathbf{K}\parallel _F^2 +L}. \end{aligned}$$

(43)

$\square$