In this paper, an impulse controller is designed for distributed parameter systems with time-varying delays, and the stability of such systems is discussed. Based on Lyapunov stability theory, sufficient conditions for the existence of exponentially stable impulse controllers for distributed parameter systems with variable time delay are obtained.
Finally, a numerical simulation is given to illustrate the effectiveness of the proposed controller. In real life and modern industrial industry, many physical systems have space-time characteristics. Their behavior must depend on time and space location, such as the process of molten steel solidification into steel plate in steelmaking plant. The space-time process of this system is called distributed parameter system. In order to study this kind of system better, quasi-linear parabolic partial differential equation (PDE) is usually constructed according to the law of conservation of energy, and the study of distributed parameter systems by using quasi-linear parabolic partial differential equation or quasi-linear parabolic partial differential-integral equation is always a phase at home and abroad.
Key research topics of scholars in related fields [1-6].
In reference [1], LUO constructs quasilinear parabolic partial differential equations for distributed parameter systems, designs controllers, and obtains sufficient conditions for the existence of exponential stability controllers for distributed parameter systems by using Lyapunov stability theorem and LMI calculation method. In [2], Wang uses Lyapunov-Krasovskii method to study the stability of parabolic partial differential equations with time delay. In reference [3], Xing studies the stability of a parabolic partial differential equation with time delay by using sliding mode control (SMC). In reference [4], Wang constructs a semi-linear parabolic differential equation for a non-linear distributed parameter system. By designing a fuzzy feedback controller combined with hybrid H2/H_ performance control, the system achieves stable state by using distributed proportional-integral space (P-Si) control.
In reference [5], Babaei constructed a semi-linear partial differential equation for a chemical distributed parameter system with unknown process parameters, and studied its adaptive output feedback control problem using Lyapunov stability theorem. In order to avoid the phenomenon of central segregation and central loosening during the solidification process of molten steel, soft reduction technology is usually used (i.e. near the solidification terminal of billet, a certain amount of pressure is applied to the billet to destroy the liquid cavity formed at the solidification terminal of billet, so as to restrain the natural production of concentrated molten steel under the action of static pressure). The technology of soft reduction is a kind of pulse control technology. In our practical engineering, how much force should be exerted and how much frequency should be used to achieve control is the key problem we need to solve.
At the same time, considering the time-delay problem caused by transmission and signal transmission, the time-delay is often variable, so it is particularly meaningful to study distributed parameter systems with variable time-delay characteristics. Based on this, this paper designs impulse controllers for distributed parameter systems with time-varying delays, and discusses the stability of such systems. By using Lyapunov stability theory, sufficient conditions for the existence of exponentially stable periodic impulse controllers for distributed parameter systems with variable delays are obtained.
Finally, combined with the given conditions, a numerical simulation is given to illustrate its effectiveness. Taking D = 2, thermostatic element A0 = 10, A = 5, and according to theorem 1A) condition, theta = 20 satisfies the condition. Assuming that the time interval of pulse control is tk-tk-1 = 0.05s, the instantaneous variable beta KF = 0.
55 describing the state can be obtained under theorem 1B condition. Assuming that, the relationship between the state and time of the system can be obtained as shown in Figure 1. Thus, under the action of the controller, the system can reach a stable state after a period of time. In this paper, an impulse controller is designed for distributed parameter systems with variable time-delay. By using Lyapunov stability theory, sufficient conditions for the existence of exponentially stable periodic impulse controllers for distributed parameter systems with variable time-delay are obtained.
Finally, a numerical simulation is given to illustrate the effectiveness of the impulse controller.