$H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917694595399680 |
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| author | Jagt, Declan S. Peet, Matthew M. |
| author_facet | Jagt, Declan S. Peet, Matthew M. |
| contents | Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) -- with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, $L_2$-gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an $H_{\infty}$-optimal estimator for 2D PDEs. In particular, we first use PIEs to represent an arbitrary well-posed 2D PDE where sensor measurements occur along some boundary of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator $\mathcal{L}$ as the observer gain. Examining the error dynamics of this estimator, it is proven that an upper bound on the $H_{\infty}$-norm of these error dynamics can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula is proposed for inversion of a class of 2D PI operators, which is then used to reconstruct the Luenberger gain $\mathcal{L}$. Results are implemented in the PIETOOLS software suite -- applying the methodology and simulating the resulting observer for an unstable 2D heat equation with boundary observations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization Jagt, Declan S. Peet, Matthew M. Optimization and Control Systems and Control Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) -- with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, $L_2$-gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an $H_{\infty}$-optimal estimator for 2D PDEs. In particular, we first use PIEs to represent an arbitrary well-posed 2D PDE where sensor measurements occur along some boundary of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator $\mathcal{L}$ as the observer gain. Examining the error dynamics of this estimator, it is proven that an upper bound on the $H_{\infty}$-norm of these error dynamics can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula is proposed for inversion of a class of 2D PI operators, which is then used to reconstruct the Luenberger gain $\mathcal{L}$. Results are implemented in the PIETOOLS software suite -- applying the methodology and simulating the resulting observer for an unstable 2D heat equation with boundary observations. |
| title | $H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2402.05061 |