Abstract
We present an optimization-based framework for analysis and control of linear parabolic partial differential equations (PDEs) with spatially varying coefficients without discretization or numerical approximation. For controller synthesis, we consider both full-state feedback and point observation (output feedback). The input occurs at the boundary (point actuation). We use positive-definite matrices to parameterize positive Lyapunov functions and polynomials to parameterize controller and observer gains. We use duality and an invertible state variable transformation to convexify the controller synthesis problem. Finally, we combine our synthesis condition with the Luenberger observer framework to express the output feedback controller synthesis problem as a set of LMI/SDP constraints. We perform an extensive set of numerical experiments to demonstrate the accuracy of the conditions and to prove the necessity of the Lyapunov structures chosen. We provide numerical and analytical comparisons with alternative approaches to control, including Sturm-Liouville theory and backstepping. Finally, we use numerical tests to show that the method retains its accuracy for alternative boundary conditions.
Original language | English (US) |
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Article number | 7517344 |
Pages (from-to) | 1636-1651 |
Number of pages | 16 |
Journal | IEEE Transactions on Automatic Control |
Volume | 62 |
Issue number | 4 |
DOIs | |
State | Published - Apr 2017 |
Keywords
- Control design
- distributed parameter systems
- partial differential equations (PDEs)
- sum of squares
ASJC Scopus subject areas
- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering