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The control of drilling vibrations: A coupled PDE-ODE modeling approach

Saldivar B., Mondié S., Ávila Vilchis J. C.

International Journal of Applied Mathematics and Computer Science

2016

Vol. 26, no. 2

335--349

The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock–bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov–Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.

Further results on robust fuzzy dynamic systems with LMI D-stability constraints

Assawinchaichote W.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 4

785--794

This paper examines the problem of designing a robust H∞ fuzzy controller with D-stability constraints for a class of nonlinear dynamic systems which is described by a Takagi–Sugeno (TS) fuzzy model. Fuzzy modelling is a multi-model approach in which simple sub-models are combined to determine the global behavior of the system. Based on a linear matrix inequality (LMI) approach, we develop a robust H∞ fuzzy controller that guarantees (i) the L2-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, and (ii) the closed-loop poles of each local system to be within a specified stability region. Sufficient conditions for the controller are given in terms of LMIs. Finally, to show the effectiveness of the designed approach, an example is provided to illustrate the use of the proposed methodology.

LMI approach to the inverse problem of optimal control.

Larin VB.

Systems Science

2000

Vol. 26, nr 3

61-68

During development of the methods of synthesis of optimal controllers (direct problem) also the inverse problem of optimal control is investigated. The algorithm for solving a direct problem is reduced to a standard procedure (construction of the stabilizing solution of a matrix Riccati equation). But, it is difficult to indicate a similar unified algorithm for inverse problems. It seems that the linear matrix inequalities (LMI) approach can be used as a basis of such a unified algorithm. Such an approach is used for developing an algorithm for solving the inverse problems.