H_\infty Optimal Estimation for Linear Coupled {PDE} Systems
In this work, we present a Linear Matrix Inequality (LMI) based method to synthesize an optimal H1 estimator for a large class of linear coupled partial differential equations (PDEs) utilizing only finite dimensional measurements. Our approach extends the newly developed framework for representing and analyzing distributed parameter systems using operators on the space of square integrable functions that are equipped with multipliers and kernels of semi-separable class. We show that by redefining the state, the PDEs can be represented using operators that embed the boundary conditions and input-output relations explicitly. The optimal estimator synthesis problem is formulated as a convex optimization subject to LMIs that require no approximation or discretization. A scalable algorithm is presented to synthesize the estimator. The algorithm is illustrated by suitable examples.
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Publication Date:
NSF-PAR ID:
10113772
Journal Name:
Proceedings of the IEEE Conference on Decision & Control
ISSN:
2576-2370
3. We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics A_t, B_t. The sequence of dynamics matrices can be arbitrary, but with a total variation, V_T, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of O(V_T^2/5 T^3/5 ). With piecewise constant dynamics, our algorithm achieves the optimal regret of O(sqrtST ) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of $V_T$. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary.more »