Optimization On Robustness Of Low-Order Uncertain Nonlinear Systems Under PID Control

In practice,a few of plants and processes can be modeled as low-order nonlinear systems,such as pendulum systems and mass-spring systems.Proportional integral derivative(PID)control is favored for its ease of implementation and undoubtedly has been the most widely and commonly used in controlling industrial processes with its simplicity and effectiveness.Therefore,the aim of this paper is to study the robustness and safety zone of PID controller parameters of low-order nonlinear systems under PID control.However,time delays are ubiquitous in various of engineering systems and industry processes,especially the bandwidth of network transmission and the driving delay of sensors.As we all known,the existence of those input delays likely degrade the system’s performance and robustness and even render a system unstable.Therefore,we consider the low-order nonlinear systems under PID control with or without time delay to analyse the robustness of systems,to find the safety zone of PID controller parameters.The main contents of this thesis are as follows:Firstly,we consider the second-order nonlinear delay systems under PID control.The systems can be transferred into a delay systems with time-varying uncertainty.We develop the stability criteria for systems and calculate the largest allowable time delay under fixed PID controller parameters.After that,an optimization algorithm based on genetic algorithm is proposed to find out the maximum allowable time delay as long as the input time delay within the range of the maximum allowable delay,there will be at least one set of PID controller parameters that can make the systems complete the robust tracking constant reference.Then,the theory of the second-order nonlinear delay systems under PID control is extended to the first-order nonlinear delay systems under PI control and PID control.The simulation results verify the feasibility and effectiveness of the theoretical results.Then,we consider the first-order nonlinear delay systems under PI control.Given the upper bound of nonlinear partial derivative and the expected allowable delay,a new algorithm combining traversal algorithm and genetic algorithm is proposed to find the safety zone of PI controller parameters,so that any group of PI controller parameters selected in the safety zone can make the first-order nonlinear delay systems achieve robust tracking constant reference.In order to reduce the time complexity of the algorithm,two new algorithms are proposed under the condition that the expected allowable delay is always changed but the upper bound of the partial derivative is invariable and the accuracy requirement of the safety zone of the PI controller parameters is not strict.Next,it is verified that when the expected allowable delay is zero,the safety zone of PI controller parameters drawn by this algorithm is the same as the previous results for the safety zone of PI controller parameters of nonlinear systems without delay.The simulation results verify the feasibility and effectiveness of the theoretical results.Lastly,we analyse the nonlinear robustness of the low-order nonlinear systems without delay.We adopt homogeneous polynomial Lyapunov function which is a new type of Lyapunov function has less conservative.We improve the previous theorem,so that the new theorem is more suitable for dealing with low-order nonlinear systems.Finally,we give the stability criterion of the systems and the method of calculating the safety zone of upper bound of partial derivatives.It is also guaranteed that the systems can achieve the robust tracking constant reference as long as the upper bound of the partial derivative of the nonlinear function belong to this region.The simulation results verify the feasibility and superiority of the theoretical results.