By Rafael Vazquez, Miroslav Krstic
This monograph offers new confident layout tools for boundary stabilization and boundary estimation for a number of sessions of benchmark difficulties in move keep watch over, with capability functions to turbulence keep watch over, climate forecasting, and plasma keep watch over. the root of the technique utilized in the paintings is the lately constructed non-stop backstepping process for parabolic partial differential equations, increasing the applicability of boundary controllers for move platforms from low Reynolds numbers to excessive Reynolds quantity conditions.Efforts in movement keep an eye on during the last few years have resulted in quite a lot of advancements in lots of assorted instructions, yet so much implimentable advancements to this point were received utilizing discretized models of the plant types and finite-dimensional regulate recommendations. by contrast, the layout equipment tested during this booklet are in response to the "continuum" model of the backstepping process, utilized to the PDE version of the movement. The postponement of spatial discretization until eventually the implementation degree deals various numerical and analytical advantages.Specific issues and contours: creation of regulate and nation estimation designs for flows that come with thermal convection and electrical conductivity, particularly, flows the place instability might be pushed via thermal gradients and exterior magnetic fields. software of a different "backstepping" process the place the boundary keep watch over layout is mixed with a specific Volterra transformation of the movement variables, which yields not just the stabilization of the circulate, but additionally the specific solvability of the closed-loop method. Presentation of a outcome unparalleled in fluid dynamics and within the research ofNavier-Stokes equations: closed-form expressions for the strategies of linearized Navier-Stokes equations less than suggestions. Extension of the backstepping method of do away with 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 study and graduate scholars within the above parts. The e-book can also be used as a supplementary textual content for graduate classes on keep watch over of distributed-parameter platforms and on movement keep an eye on.
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Additional info for Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation
1 is shown to hold for the inverse kernel. 4 Singular Pertubation Stability Analysis for the System Now that we have derived a control law for the reduced system, we can drop the assumption that ǫ = 0 and instead consider it a small but nonzero parameter, and analyze the stability of the closed-loop system. Now the quasi-steady-state solution is no longer the exact solution of the v PDE, but it still plays an important role. 34) an error variable z that measures the deviation of the velocity from the fast solution can be introduced: z(t, r) = v(t, r) − vss (t, r).
6 Notes and References In this chapter we have considered a 2D model of thermal-fluid convection that exhibits the prototypical Rayleigh–Bernard convective instability . Stabilizing controllers have been designed for this problem in the past, including an LQG controller by Burns et al. , who formulated the problem, and a nonlinear backstepping design for a discretized version of the plant . The design in this chapter is simpler than the former, not needing a solution of Ricatti equations, only a linear hyperbolic equation; and more rigorous than the latter, which does not hold in the limit when the discrete grid approaches the continuous domain.
47) R1 where β1 = √ Q2wz 2π ∞ + (R22 − R12 ) ln R2 1 Q R1 wz ∞ . 52) ∞, 2 ∞ R2 , 2 ∞. 55) In both of the previous calculations, repeated use of Cauchy–Schwartz’s and Young’s inequality has been made, and the following lemma (a version of Poincar´e’s inequality) has been employed. 1 For any τ ∈ H 1 ((R1 , R2 ), L2 (0, 2π)), the following inequality holds: 2π 0 R2 τ 2 (r, θ)rdrdθ R1 2π ≤ 2R2 (R2 − R1 ) τ 2 (R2 , θ)dθ 0 2π + 4(R2 − R1 )2 0 R2 R1 τr2 (r, θ)rdrdθ. 56) Proof. 58) which, integrated in angle from 0 to 2π, yields the result.
Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation by Rafael Vazquez, Miroslav Krstic