Constrained Feedback Stabilization for Bilinear Parabolic Systems

Abstract

In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.

Tracking control for planar nonminimum-phase bilinear control system with disturbance using backstepping

“…This article presents the control design of a non-minimum phase bilinear control system containing a disturbance function. The main difference with previous studies in [12]- [14], [16], [18] is in terms of the presence of a disturbance function in the system and the type of system and in [15], [34]- [36] is in terms of the use of the control design method used. The method used in this article is the backstepping method, which develops the previous results in [37].…”

“…Because max(𝑑) > 𝑑(𝑡) then sign(𝑤)𝑑(𝑡) − max (𝑑) < 0 and 𝑉 ̇2(𝑡) < 0 apply to every 𝑡 ≥ 0. Furthermore, the variable control 𝜈(𝑡) is obtained from (15) and using the relation between the state variables {𝑒 1 (𝑡) , 𝑒 2 (𝑡) , 𝑤(𝑡)} in (12), the following control function is obtained (18).…”

“…Using Table 1, the control function in (18), which takes the output to the origin, is 𝜈(𝑡) = −1.3103(17.5910𝑥 2 − 11.1970𝑤 − 0.7632 sign(𝑤)). Figure 1 shows the simulation results for the stabilisation problem.…”

“…Some examples of the use of other bilinear control systems can be seen in the boost converter problem in [11], [12], on controlling the spread of disease in [13], [14], on directing ship motion in [15], on regulating hot water storage in [16], on the diesel engine fuel collector in [17]. The application of control to the Hilbert chamber using finite feedback has also been carried out [18]. The model's accuracy is also determined from the modelling process carried out.…”