Stabilization of a Heat‐ODE System Cascaded at a Boundary Point and an Intermediate Point

Abstract

This paper considers the stabilization of a heat‐ODE system cascaded at a boundary point and an intermediate point. The stabilizing feedback control law is designed by the backstepping method. Based on a novel transformation, we prove that all the kernel functions in the forward and inverse transformations are of the class C2. Moreover, the effectiveness of controller design is shown with a numerical simulation. Finally, we show the coherence between the controllability assumption of the main theorem in this paper and the known one for a special case with λ=0.

Control of two Electrical Plants

“…Let: A i and B i be described in (6) and A ci be described in (7), j be the eigenvalues of A ci , and Re j be the real parts of the eigenvalues j . Then 1.-The closed loop model equations (7) are uniformly stable if Re j < 0 for all eigenvalues of A ci . 2.-The closed loop model equations (7) are unstable if Re j > 0 for one or more of the eigenvalues of A ci .…”

“…In [1] and [2], mathematical models of the gas turbines are described. In [5][6][7], feedback linearization techniques are employed. In [5][6][7], feedback linearization techniques are employed.…”

“…Optimal controls are addressed in and . In , feedback linearization techniques are employed. Strategies for power control are focused in and .…”

Stabilization of a wave equation with a distributed control

“…Many systems can be described by partial differential equations (PDEs) in engineering fields, and stabilizing these systems is an important and basic problem; therefore, in control theory field, the research of stabilization is very popular and basic, and there are many methods involved in, for example, backstepping method [1], spectral methods [2], linear quadratic regulator (LQR) approach [3], multiplier technique [4], and Lyapunov function method [5]. Among these methods, the backstepping method is a high efficient tool in designing controller for boundary controls of various types PDEs (see previous studies [6][7][8][9][10][11][12][13][14][15] and the references therein), because this method is easy to understand and it does not need the advanced mathematical tools; therefore, it is very useful in designing the stabilization controller of system.…”

Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient

“…Also, the authors of [15] designed a backstepping controller based observer for one dimensional linear parabolic PDEs. Then, in [16] the exponential stabilization of a cascaded system with a heat equation and an ordinary differential equation is studied. A backstepping observer is applied for linear PDEs on higher dimensional spatial domains in [17].…”