A Closed-Form Observer for the 3D Inductionless MHD and Navier-Stokes Channel Flow

Vázquez, Rafael; Schuster, Eugenio; Krstić, Miroslav

doi:10.1109/cdc.2006.376691

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…Analysis and control of Navier-Stokes equations [1,7,13,18,20,19,24,37,39,59,60,61,64,66,72,75,78,142,146,162,163,165,170,171,172,173,180,179] are exciting areas attracting the attention of those who seek some of the ultimate challenges in applied mathematics and control theory. A detailed overview of various efforts in controllability, optimal control, stabilization, and boundary control of Navier-Stokes is beyond the scope of this introductory book; however, we point out that good surveys reflecting the state of the art at the time of their publication are contained in [1,24,64].…”

Section: Notes and Referencesmentioning

confidence: 99%

Boundary Control of PDEs

Krstić

Smyshlyaev

2008

Self Cite

1,044

248

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Backstepping, on the other hand, requires little background beyond calculus for users to understand the design and the stability analysis. This book is designed to be used in a one semester course on backstepping techniques for boundary control of PDEs. In Fall we offered such a course at the University of California, San Diego. The course attracted a large group of postgraduate, graduate, and advanced undergraduate students. Due to the diversity of backgrounds of the students in the class, we developed the course in a broad way so that students could exercise their intuition no matter their backgrounds, whether in fluids, flexible structures, heat transfer, or control engineering. The course was a success and at the end of the quarter two of the students, Matthew Graham and Charles Kinney, surprised us with a gift of a typed version of the notes that they took in the course. We decided to turn these notes into a textbook, with Matt's and Charles' notes as a starting point, and with the homework sets used during the course as a basis for the exercise sections in the textbook. We have kept the book short and as close as possible to the original course so that the material can be covered in one semester or quarter. Although short, the book covers a very broad set of topics, including most major classes of PDEs. We present the development of backstepping controllers for parabolic PDEs, hyperbolic PDEs, beam models, transport equations, systems with actuator delay, Kuramoto-Sivashinsky-like and Korteweg-de Vries-like linear PDEs, and Navier-Stokes equations. We also cover the basics of motion planning and parameter-adaptive control for PDEs, as well as observer design with boundary sensing. Short versions of a course on boundary control, based on preliminary versions of the book, were taught at the

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Section: Notes and Referencesmentioning

confidence: 99%

Boundary Control of PDEs

Krstić

Smyshlyaev

2008

Self Cite

1,044

248

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“…One of the advantages of the backstepping approach over optimal control approaches, when applied to the channel flow, is that it is not necessary to solve high-dimensional Riccati equations, and the backstepping gains (the kernels) are explicit (symbolically computable) functions of the Reynolds number and the wavenumbers. Though the feedback law that we present here employs full state feedback, an observer developed in Vazquez, Schuster & Krstic (2006b) allows us to implement the controller by measuring only the pressure and the skin friction at the same wall as the output reference y = 0.…”

Section: Stabilizationmentioning

confidence: 99%

Motion planning and trajectory tracking for three-dimensional Poiseuille flow

Cochran

Krstić

2009

J. Fluid Mech.

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We present the first solution to a boundary motion planning problem for the Navier–Stokes equations, linearized around the parabolic equilibrium in a three-dimensional channel flow. The pressure and skin friction at one wall are chosen as the reference outputs as they are the most readily measurable ‘wall-restricted’ quantities in experimental fluid dynamics and also because they play a special role as performance metrics in aerodynamics. The reference velocity input is applied at the opposite wall. We find the exact (method independent) solution to the motion planning problem using the PDE (partial differential equation) backstepping theory. The motion planning solution results in open-loop controls, which produce the reference output trajectories only under special initial conditions for the flow velocity field. To achieve convergence to the reference trajectory from other (nearby) initial conditions, we design a feedback controller. We also present a detailed examination of the closed-form solutions for gains and the behaviour of the motion planning solution as the wavenumbers grow or the Reynolds number grows. Numerical results are shown for the motion planning problem.

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“…But in many applications such as earth sciences or engineering, the systems are modeled using PDE that are highly nonlinear and, in many instances, chaotic. In such cases of infinite dimensional systems governed by PDE, there are only a few examples available in the literature, mostly for linear systems and very few for nonlinear systems [23,31,36,12,16,17,20,35,26,6,30].…”

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confidence: 99%

“…There has been some work on developing observers for Navier-Stokes based systems. The papers [35,20,28] work with finite dimensional approximations of the PDE involved whereas we work with the full PDE itself. A completely different approach based on developing observers using appropriate continuous time limits of discrete time 3D-var or Kalman filter is developed in a series of papers [22,24,8] and also in [7,2,1,27,18,4].…”

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Observers for Compressible Navier--Stokes Equation

Apte¹,

Auroux²,

Ramaswamy³

2018

SIAM J. Control Optim.

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We consider a multi-dimensional model of a compressible fluid in a bounded domain. We want to estimate the density and velocity of the fluid, based on the observations for only velocity. We build an observer exploiting the symmetries of the fluid dynamics laws. Our main result is that for the linearised system with full observations of the velocity field, we can find an observer which converges to the true state of the system at any desired convergence rate for finitely many but arbitrarily large number of Fourier modes. Our one-dimensional numerical results corroborate the results for the linearised, fully observed system, and also show similar convergence for the full nonlinear system and also for the case when the velocity field is observed only over a subdomain.

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A Closed-Form Observer for the 3D Inductionless MHD and Navier-Stokes Channel Flow

Cited by 7 publications

References 25 publications

Boundary Control of PDEs

Boundary Control of PDEs

Motion planning and trajectory tracking for three-dimensional Poiseuille flow

Observers for Compressible Navier--Stokes Equation

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