Stability And Control Problems For Second Order Delay Differential Systems

Delay phenomenon is a very common phenomenon in nature.It is widely used in vehicle active suspension system,rocket engine combustion dynamics and other mechan-ical systems.The delay phenomenon makes the evolution of the system state over time dependent not only on the current state of the system,but also on the past period of time state of the system.Delay systems are the main mathematical models with delay effects in engineering problems.In this Master's thesis,the finite time stability and controllability of second order delay differential systems are analyzed,and the Iterative Learning Control technique is used to realize the finite time tracking control for second order delay differential systems.Firstly,we study the finite time stability of the system.By using the power series expansion of hyperbolic sine/cosine function,we obtain the norm estimation of the delay matrix sine/cosine function corresponding to the second order delay differential equation.Three sufficient conditions for the finite time stability of the system are given,and a numerical examples is used to verify the validity of the theoretical results.Secondly,we study the controllability of the system,construct the delayed Gramian matrix by using the delay matrix sine function,then establish the criterion of the con-trollable delay Gramian matrix of the system,give the concrete calculation method of the control function,and generalize the classical results of the linear system.Finally,we study the Iterative Learning Control problem of the system.For the second order characteristic and different output of the system,the open loop,closed loop and open closed loop iterative learning laws of typeand~2are designed,respectively,and the convergence of the system output is analyzed.The effectiveness of the iterative learning algorithm is verified by a numerical simulation example,and the exact error of each iteration is given.