Book Description

As a significant class of nonlinear systems, bilinear systems (BLS) are extensively developed in the past years. In addition to their advantages over linear systems which are often not adequate to represent accurately many control processes, the BLS are particularly appealing in modeling biological systems in which parametric controls are of fundamental importance. This dissertation is aimed at making a contribution to the theory of bilinear control systems via Lie algebraic methods, and the application of accomplished results in immunology. To this end, the input-output relationship of BLS is studied in terms of Volterra series expansion which provides a convenient tool in examining the BLS stability as well as controllability. Moreover, the Volterra series associated with a BLS is in general an infinite series, and thus in practice, it is important to know how many terms are required to prevent the truncation error from exceeding the maximum allowance. Criteria of acquiring prescribed accuracy in terms of finite Volterra series are derived for BLS with uniformly bounded input or with exponentially stable linear subsystems. The problem of inverse system design which is capable of identifying both the input function and the state variables based upon the output data, is also considered. The observer theory of constant linear systems is then extended into a special class of bilinear systems with input matrices of rank one. In view of the functional similarity between immune processes and parametric control systems, a mathematical model of humoral immune response is presented and analyzed. The structural aspects of this nonlinear immune model is examined with the aid of bilinear control theory. Approximate immune models which are amenable to the control-theoretic analysis via foregoingly developed techniques are proposed. Some computer simulations are performed to show that the form of model responses is reasonable by comparing with the experimental data.