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Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed $L^2$-Gain
- Source :
- IEEE Transactions on Automatic Control. 67:5570-5577
- Publication Year :
- 2022
- Publisher :
- Institute of Electrical and Electronics Engineers (IEEE), 2022.
-
Abstract
- Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under the assumption that one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers for 1D parabolic PDEs with both (observation and control) operators being unbounded. We consider the Kuramoto-Sivashinsky equation (KSE) under either boundary or in-domain point measurement and boundary actuation. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension $N$ may be larger than this number. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in an LMI whose elements and dimension depend on $N$. The value of $N$ and the decay rate are obtained from the LMI. We extend our approach to internal stabilization with guaranteed $L^2$-gain and input-to-state stabilization. We prove two crucial properties of the derived LMIs. First, We prove that the LMIs are always feasible provided $N$ and the $L^2$ or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some $N$ implies its feasibility for $N+1$ (i.e., enlarging $N$ in the LMI cannot deteriorate the resulting decay rate of the closed-loop system). Numerical examples demonstrate the efficiency of the method.
- Subjects :
- Observer (quantum physics)
Boundary (topology)
Eigenfunction
Computer Science Applications
Operator (computer programming)
Dimension (vector space)
Optimization and Control (math.OC)
Control and Systems Engineering
Control theory
Bounded function
FOS: Mathematics
Applied mathematics
Heat equation
Electrical and Electronic Engineering
Mathematics - Optimization and Control
Mathematics
Subjects
Details
- ISSN :
- 23343303 and 00189286
- Volume :
- 67
- Database :
- OpenAIRE
- Journal :
- IEEE Transactions on Automatic Control
- Accession number :
- edsair.doi.dedup.....94191c31b0eb3cd4fd7db862495fb003