Controllability And Stabilization Of Bilinear Systems

Bilinear systems are an important class of nonlinear systems, which are capable of repre-senting a wide range of physical, chemical, biological, and social system processes. This kind of systems has a special feature that it is linear with respect to state or control, but not jointly linear in state and control, because it contains the cross product of state and control. As a sim-pler class of nonlinear systems, bilinear systems can well approximate a wide class of nonlinear control systems, and there is certain degree of similarities between bilinear systems and linear systems in structure, which make bilinear systems have wide applications in engineering fields and is of great importance in theory. In the past few decades, the research on bilinear systems is one of hotspot in control field as well as mathematics, due to receiving much attention in term of many engineering applications and important theoretical value.In this dissertation, we mainly study controllability and stabilization of bilinear systems, and obtain some important theoretical and practical results. The main contents and contribution of this dissertation are summarized as follows.1. Controllability of a class of two-input, two-dimensional, discrete-time homogeneous bilinear systems is discussed. Supposing the two coefficient matrices are commutative each other, we classify these systems and discuss each case. Some sufficient and necessary conditions are presented for each of them. Thus the controllability problem of these systems is completely solved by assuming that the three coefficient matrices are commutative each other. It has clearly shown that the controllability of some bilinear systems may or may not change after Euler discretization.2. We study the controllable region of a class of single-input, n-dimensional discrete-time bilinear systems that is uncontrollable. In theory, the bilinear system, whose dimension is higher than two, has been proved to be not controllable. But we can analysis how many uncontrollable points there are in the whole space. When the coefficient matrix is cyclic and the order of the largest Jordan block is not more than two in its Jordan canonical form, it will be shown that the uncontrollable region of this kind of bilinear systems have Lebesgue measure zero. In other word, the kind of bilinear systems is controllable almost everywhere in the whole space. Besides, calculations demonstrate that if the initial state is taken from the controllable region, then it can reach any point of the whole space.3. We study the stabilization of a class of continuous bilinear systems with multiple inputs. Based on Lyapunov stability theory, the stabilization of these bilinear systems is investigated in cases of having a drift and having no drift, respectively. Some explicit state feedback laws are designed. Theoretically we have proved that a bilinear system is globally asymptotically stable under these control laws, which shows that bilinear systems can be stabilized globally. By doing numerical simulation for two-dimensional and three-dimensional bilinear systems, respectively, we can intuitively see that, with the increase of time t, the trajectories of a bilinear system are close to the origin gradually. The simulating results demonstrate that our results are correct.4. We study the stabilization of discrete bilinear systems which are got by Euler discretiza-tion of continuous homogenous bilinear systems. Two classes of feedback laws are proposed, which are state feedback and constant feedback respectively. In special, when satisfying some conditions, these discrete bilinear systems are not only asymptotically stable, but also expo-nentially stable by state feedback laws. For the results, some simulations are did, and we can easily see that, with the increase of the transform, the trajectories of the discrete bilinear system will converge to the origin gradually. So the simulating results are well consistent with the our theoretical results.