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 accuracy of the conditions and to prove 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. Index Terms-Distributed parameter systems, partial differential equations (PDEs), control design, sum of squares.
In this paper, we propose a method to synthesize infinite-dimensional observer-based controllers for partialdifferential systems. To illustrate the approach, we use a onedimensional model of heat conduction with point observation and boundary control. Our method uses Sum-of-Squares optimization to solve linear operator inequalities in an infinite-dimensional Hilbert space. We use the semigroup framework and an operator version of the separation theorem and the Lyapunov inequality. We implement our method using SOSTOOLS and SeDuMi. We simulate the effects of our controller using Matlab.
In this paper we present a Lyapunov based feedback design strategy, by employing the sum-of-squares polynomials framework, to maximize the bootstrap current in tokamaks. The bootstrap current may play an important role in reducing the external energy input required for tokamak operation. The sum-of-squares polynomials framework allows us to algorithmically construct controllers. Additionally, we provide a heuristic to take into account the control input shape constraints which arise due to limitations on the actuators.
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