On Discretization Of Sliding Mode Control For Linear Systems

Sliding mode control is also called variable structure control, which is a kind of special nonlinear discontinuous control firstly proposed by Soviet researcher Emelyanov in1950s. Furthermore, this theory was greatly developed by Utkin and Itkis. Since sliding mode control is robust to uncertain parameters and ex-ternal disturbances, it has been paid great attention by lots of researchers in the control community and applied in many practical systems. With more and more applications of computer control technique, the discrete control theory based on digital computer control has been attached great importance. For the convenience of applying sliding mode control technique, it is very necessary to investigate the discrete forms of the continuous sliding model control laws. There have been lots of results on discrete sliding mode control. But it has been undertaken mainly in two streams. One is the discrete-time design approach where the controlled systems are discretized first and a discrete sliding mode control law is then de-signed by the generalized concepts and results from the continuous control theo-ry. The other is directly imposing the discrete forms of continuous sliding mode controllers to the computer controlled systems. For the later, the existing result-s show that there exist complex dynamical behaviors such as multi-periodical orbits and chaos in discretized sliding mode control systems. Thus, the theoret-ical analysis for the systems controlled by the discretized sliding mode control is very challenging. There are still many deficiencies and unsolved problems in discretized sliding mode control systems. For example, some a priori assumption-s imposed on the states of the closed-loop systems are often used in analyzing the dynamical behavior. Moreover, how to improve the existing discrete sliding mode controller and how to reveal the relationship between the dynamical be- haviors and the sampling period are also worthy to be investigated further. The obtained results in this dissertation just focus on the second issue and are listed as follows.1. A correction term is introduced in the discrete form of the continuous time sliding mode control law for eliminating the complex dynamical behavior of the closed-loop system.2. It is proved that the trajectories of the closed-loop systems are uniformly bounded respect to sufficient small sampling periods and converge to2-periodic orbits with1-order infinitesimal boundaries with respec-t to the sampling period.3. For multi-input high-order linear systems, a new system transformation is proposed so that the obtained results for the second or-der systems can be generalized to multi-input high-order systems.4. For the double integrator systems, an analytical expression of the limit periodic orbits is obtained.