Study Of The Useness Of Delta Operator In Control Theory

The development of discrete sampling system has been increasing for rapid progress of the computer technique. Most traditional digital signal processing and control algorithms are inherently ill-conditioned when applied in situations in which data are taken at sampling rates that are high related to the dynamics of the underlying continuous-time processes being sampled. Considerable recent progress towards ameliorating such ill-conditioning problem has been made through the use of a divided-difference operator, rather than the conventional shift operator, to represent the dynamics of sampled data. The delta operator enables a smooth transition of the sampled data to their continuous-time counterparts as a unified method of the continuous-time system and discrete-time system.We introduced firstly the fundamental theories. Based on the existent results, we deduced and proved some other results, such as delta operator transition, anti-transition and several characteristical theories of the delta transition, the mappings of the s, z and 8 domains are also considered.We discussed the method of state space design, and deduced the Ackermann formulation. In the end, separation theory is adapted to the discrete system of the delta domain.In this paper, we used the characteristics and introduced the delta operator to linear quadratic following control system, the results show that when sampling period is approaching to zero the results of the discrete-time system is approaching the continuous-time system, and the design of the following controller is completed by linear Riccati equation.The delta operator is introduced to the H9 filter problem of discrete system, which is significant for the anti-disturbance of the high-speed sampled system. In the delta domain system, the coefficient of the gain matrix of the estimator is approaching to that of the continuous-time system. Which not only guarantees the stability and the performance of the system but also avoids thealso avoids the defects the z domain when the sampling periods is approaching to zero.For the delta domain systems with the discrete systems, we analyze and design the Hx controllers using Lyapunov stability theory. The results contain the sampling period and show that when sampling period is decreasing to zero the system is stable all the same while the performance of the discrete-time system is approaching to the continuous-time system.To study the realizition of the delta operator,we simulated and obtained a series of figures using Matlab.From these figrues it can be seen that the delta form has clearly superior numerical properties ralative to the shift form.