Robust H_∞ Controller for Bilinear System to Minimize HIV Concentration in Blood Plasma

Abstract

Human Immunodeficiency Virus (HIV) is a type of virus which attacks CD4+ T cells. Insufficient numbers of CD4+ T cells will affect the performance of immunity systems so that someone become more risky to have AIDS or other diseases. HIV phenomenon is modelled as nonlinear system with disturbance, but there is no exact method to solve problems that related with analyzing nonlinear systems with control treatment. Thus, the nonlinear system is approximated into a bilinear system by using Carleman Bilinearization method. A robust H ∞ controller is designed as a control input to accomodate the disturbances in the dynamic system. Generally, the treatment for HIV is merely to obstruct the replication process. Therefore, the robust control input is described as the chemotherapy of hypothetical drugs which aimed to clear off the HIV concentration in blood plasma. Robust H ∞ control is built with the coefficient matrix of the bilinear system. The simulation result is analyzed by comparing the effect of the robust control toward bilinear and nonlinear system. From numerical simulations, an individual will experience heavy symptomps if the hypothetical drugs chemotherapy is not given. This finding is strengthening the urgency of robust control as a chemotherapy with a waiting time. Based on the numerical simulation with robust H ∞ control input, bilinear system has trends that give similar interpretation with the trends of nonlinear system. Besides that, robust H ∞ control on bilinear system has demolishing speed of HIV concentration and infected CD4+ T cells concentration which approximately equal to the demolishing speed from nonlinear system. Therefore, we can conclude that the performance of H ∞ robust control on bilinear system is nearly same as the performance of nonlinear system.

Structure-preserving interpolation of bilinear control systems

“…where X(s), U (s), and Y (s) denote the Laplace transforms of the time-dependent functions x(t), u(t), and y(t). Inspired by much richer structured systems than (6) appearing in the linear case such as those describing the dynamic response of a viscoelastic body, [26] introduced a more general system of equations in the frequency domain, given by…”

“…In the last decades, the class of bilinear systems became an essential tool in systems theory. They naturally appear in the modeling process of many physical phenomena, e.g., in the modeling of population, economical, thermal, and mechanical dynamics [1,2], of electrical circuits [3], of plasma devices [4,5], or of medical processes [6]. Bilinear systems can also result from approximation of general nonlinear systems employing the Carleman linearization process [7,8].…”

Structure-preserving interpolation for model reduction of parametric bilinear systems

“…Design and control processes usually involve simulating systems of differential equations describing the underlying dynamics. In the setting of nonlinear or stochastic processes, an important class of such systems are parametric bilinear time-invariant systems; see, e.g., [1,14,15,18] for some applications of bilinear systems. In most cases, these bilinear systems have special structures resulting from the underlying physical model and the dynamics are parameter dependent.…”

Structure-Preserving Interpolation of Bilinear Control Systems

“…In the last decades, the class of bilinear systems became an essential tool in systems theory. They naturally appear in the modeling process of many physical phenomena, e.g., in the modeling of population, economical, thermal and mechanical dynamics [28,29], of electrical circuits [2], of plasma devices [30,31], or of medical processes [34]. Bilinear systems can also result from approximation of general nonlinear systems employing the Carleman linearization process [16,26].…”