Control of Turbulent and Magnetohydrodynamic Channel Flows: by Rafael Vazquez, Miroslav Krstic
By Rafael Vazquez, Miroslav Krstic
This monograph provides new positive layout equipment for boundary stabilization and boundary estimation for numerous periods of benchmark difficulties in circulation regulate, with capability purposes to turbulence keep watch over, climate forecasting, and plasma keep watch over. the foundation of the strategy utilized in the paintings is the lately constructed non-stop backstepping technique for parabolic partial differential equations, increasing the applicability of boundary controllers for movement structures from low Reynolds numbers to excessive Reynolds quantity conditions.
Efforts in movement regulate during the last few years have ended in a variety of advancements in lots of varied instructions, yet such a lot implementable advancements to this point were received utilizing discretized models of the plant types and finite-dimensional keep an eye on innovations. against this, the layout tools tested during this publication are according to the “continuum” model of the backstepping procedure, utilized to the PDE version of the circulation. The postponement of spatial discretization until eventually the implementation degree bargains a variety of numerical and analytical advantages.
Specific issues and features:
* creation of keep an eye on and kingdom estimation designs for flows that come with thermal convection and electrical conductivity, specifically, flows the place instability will be pushed via thermal gradients and exterior magnetic fields.
* program of a distinct "backstepping" strategy the place the boundary keep an eye on layout is mixed with a specific Volterra transformation of the movement variables, which yields not just the stabilization of the stream, but in addition the specific solvability of the closed-loop system.
* Presentation of a consequence unheard of in fluid dynamics and within the research of Navier–Stokes equations: closed-form expressions for the strategies of linearized Navier–Stokes equations lower than feedback.
* Extension of the backstepping method of cast off one of many well-recognized root explanations of transition to turbulence: the decoupling of the Orr–Sommerfeld and Squire systems.
Control of Turbulent and Magnetohydrodynamic Channel Flows is a superb reference for a extensive, interdisciplinary engineering and arithmetic viewers: keep watch over theorists, fluid mechanicists, mechanical engineers, aerospace engineers, chemical engineers, electric engineers, utilized mathematicians, in addition to learn and graduate scholars within the above components. The publication can also be used as a supplementary textual content for graduate classes on regulate of distributed-parameter platforms and on circulation control.
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Extra info for Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation
The simplicity of ﬁnding the gains is one of the main beneﬁts of the backstepping method. The standard methods for control of PDE, which are “PDE extensions” of linear-quadratic-optimal (LQR) methods for ﬁnite-dimensional systems, obtain their gains through the solution of Riccati equations—quadratic operator-valued equations, which are, in general, hard to solve. However, the numerical advantages that backstepping oﬀers are just one of its appeals. The other is the conceptual elegance of using a feedback transformation that eliminates exactly the part of the PDE that is undesirable (or adds a part that is missing in a PDE, like damping), while leaving the PDE in a familiar form where physical intuition can be used in shaping the closed-loop dynamics.
133) for G can be integrated, yielding as integral equation for G G(δ, η) = −g(η) − η + 0 + 1 4 δ 1 4 1 τ dτ − 2 4 λ η G (η + τ, η − τ ) g(τ )dτ + δ η μ+η−s G(τ, s)f η 0 μ δ s+τ s−τ , 2 2 η f 0 η δ 1 4 η λ 0 η τ −s 2 τ −s τ +s ,μ− 2 2 dτ ds G(τ, s)dsdτ dτ dsdμ. 136). 2 Mathematical Preliminaries and Notation 27 and for n ≥ 1, η Gn (δ, η) = Gn−1 (δ, η) + + + 1 4 1 4 δ η δ μ+η−s μ η λ η Gn−1 (η + τ, η − τ ) g(τ )dτ Gn−1 (τ, s)f 0 η 0 0 τ −s 2 τ +s τ −s ,μ− 2 2 Gn−1 (τ, s)dsdτ. 138) Then it can be shown that G(δ, η) = lim Gn (δ, η).
5 Simulation Study 51 This establishes asymptotic stability for the plant in the z-, w-coordinates, when ∈ (0, ∗ ). 67) which are derived taking the norm in the respective deﬁnitions. 29) is exponentially stable at the origin in the L2 sense, that is, there exist positive constants M and α, independent of the initial conditions, such that R2 2π v 2 (t, s) + ≤ M e−αt τ 2 (t, s, φ)dφ sds 0 R1 R2 2π v 2 (0, s) + R1 τ 2 (0, s, φ)dφ sds. 8) and the form of the boundary conditions. 5 Simulation Study We show a prototypical simulation case.