This paper revisits the design of compensator-based controller for a class of infinite dimensional systems. In order to save computational time, a functional observer is employed to reconstruct a functional of the state which coincides with the full state feedback control signal. Such a full-state feedback corresponds to an idealized case wherein the state is available. Instead of reconstructing the entire state via a state-observer and then use this state estimate in a controller expression, a functional observer is used to estimate the product of the state and the feedback operator, thus resulting in a significant reduction in computational load. This observer design is subsequently integrated with a sensor selection in order to improve controller performance. An appropriate metric is used to optimize the sensor location resulting in improved performance of the functional observer-based compensator. The integrated design is further extended to include a controller with an unknown input functional observer. The results are applied to 2D partial differential equations and detailed numerical studies are included to provide an appreciation in the significant savings in both operational and computational costs.
Finite Dimensional Functional Observer Design for Parabolic Systems
This paper combines two control design aspects for a class of infinite dimensional systems, and each of the designs aims at significantly reducing the implementation complexity and computational load. A functional observer, and its extension of an unknown input functional observer, aims to reconstruct a functional of the infinite dimensional state. The resulting compensator only requires the solution to an operator Sylvester equation plus one differential equation for each dimension of the control signal, as opposed to an infinite dimensional filter evolution equation and an associated operator Riccati equation for the filter operator covariance. When the functional to be estimated coincides with the expression of a full state feedback control signal, then the functional observer becomes the minimum order compensator. When the parabolic system admits a decomposition whereby the system is decomposed into a lower finite dimensional subspace comprising the unstable eigenspectrum and an infinite stable subspace, then the functional observer-based compensator design becomes the minimum order compensator for the finite dimensional subsystem. This approach dramatically reduces the computation for solving the ARE needed for the full state controller and the associated Sylvester equation needed for the functional observer. Numerical results for a parabolic PDE in one and two spatial more »
- Award ID(s):
- 1825546
- Publication Date:
- NSF-PAR ID:
- 10385851
- Journal Name:
- 2021 60th IEEE Conference on Decision and Control
- Page Range or eLocation-ID:
- 1155 to 1160
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This paper extends earlier results on the adaptive estimation of nonlinear terms in finite dimensional systems utilizing a reproducing kernel Hilbert space to a class of positive real infinite dimensional systems. The simplest class of strictly positive real infinite dimensional systems has collocated input and output operators with the state operator being the generator of an exponentially stable C 0 semigroup on the state space X . The parametrization of the nonlinear term is considered in a reproducing kernel Hilbert space Q and together with the adaptive observer, results in an evolution system considered in X × Q. Using Lyapunov-redesign methods, the adaptive laws for the parameter estimates are derived and the well-posedness of the resulting evolution error system is summarized. The adaptive estimate of the unknown nonlinearity is subsequently used to compensate for the nonlinearity. A special case of finite dimensional systems with an embedded reproducing kernel Hilbert space to handle the nonlinear term is also considered and the convergence results are summarized. A numerical example on a one-dimensional diffusion equation is considered.
-
The Sylvester equation offers a powerful and unifying primitive for a variety of important graph mining tasks, including network alignment, graph kernel, node similarity, subgraph matching, etc. A major bottleneck of Sylvester equation lies in its high computational complexity. Despite tremendous effort, state-of-the-art methods still require a complexity that is at least \em quadratic in the number of nodes of graphs, even with approximations. In this paper, we propose a family of Krylov subspace based algorithms (\fasten) to speed up and scale up the computation of Sylvester equation for graph mining. The key idea of the proposed methods is to project the original equivalent linear system onto a Kronecker Krylov subspace. We further exploit (1) the implicit representation of the solution matrix as well as the associated computation, and (2) the decomposition of the original Sylvester equation into a set of inter-correlated Sylvester equations of smaller size. The proposed algorithms bear two distinctive features. First, they provide the \em exact solutions without any approximation error. Second, they significantly reduce the time and space complexity for solving Sylvester equation, with two of the proposed algorithms having a \em linear complexity in both time and space. Experimental evaluations on a diverse setmore »
-
This paper presents an adaptive functional estimation scheme for the fault detection and diagnosis of nonlinear faults in positive real infinite dimensional systems. The system is assumed to satisfy a positive realness condition and the fault, taking the form of a nonlinear function of the output, is assumed to enter the system at an unknown time. The proposed detection and diagnostic observer utilizes a Reproducing Kernel Hilbert Space as the parameter space and via a Lyapunov redesign approach, the learning scheme for the unknown functional is used for the detection of the fault occurrence, the diagnosis of the fault and finally its accommodation via an adaptive control reconfiguration. Results on parabolic PDEs with either boundary or in-domain actuation and sensing are included.
-
This work considers the replacement of a full-state feedback controller by a static output feedback controller employing a finite number of point sensors. This is achieved by the approximation of the feedback kernel associated with the full state feedback operator. The feedback kernel is partitioned into equiareal cells and an appropriately selected centroid within each cell serves as the sensor location. This allows one to approximate the inner product of the feedback kernel and the full state by the finite weighted sum of static output feedback measurements. By equating the feedback kernel with the density of a hypothetical sensor network, the problem of approximating the sensor density becomes that of partitioning the sensor density using the proposed computational-geometry based decomposition that is based on a modification of Centroidal Voronoi Tessellations. When the control is considered over a finite horizon and/or the actuator itself is repositioned within the spatial domain, the resulting feedback kernel is rendered time-varying. This requires its partitioning at each time leading to mobile sensors within the spatial domain. Two guidance policies are proposed: one uses the partitioning of the kernel method at each time to find the optimal sensors thus resulting in moving sensors. The other methodmore »
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Publisher's Version of Record
- https://doi.org/10.1109/CDC45484.2021.9682848
