Design Of Time-delayt Controllers For Arbitrary-order Linear Systems

In recent years, the study of time delay demonstrates that the time delay can be beneficial or detrimental to the stability and performance of a control system. A stable system becomes unstable after introducing a time delay. On the contrary, introducing a time delay can stabilize an unstable system. Because of the important role that the time delay plays and the utility of different types of time delay in control systems and biological networks, time-delay controllers can be composed of different time-delay modules. The design of time-delay controllers has not been solved very well. Thus, the design of time-delay controllers for arbitrary-order linear systems is studied in this thesis. The methods of solving the stabilizing region of control parameters are proposed. The values which are in the stabilizing region can stabilize the systems.For the continuous systems, the stabilizing region of the single-delay controller is found based on the Hurwitz stability condition and Hermite-Biehler theorem. For two-delay controllers,the stabilizing region of parameters is solved by treating the characteristic polynomial as the form of vectors addition and constructing a triangle. Also, based on the stabilizing region of time-delay controllers, a method of designing H?three-delay controller for systems with time delay is proposed. The problem of finding the set of H?three-delay parameters satisfying the certain H?performance is converted to the problem of stabilizing the quasi-polynomials and the characteristic functions which can be solved by using the D-composition method. Then, the intersection of the control parameters that can make the system stable and satisfy the H?performance index is the set that the H?three-delay control parameters satisfy the requisite H?performance.For the discrete systems, the problem of stabilizing the system with two-delay controller is solved by computing the boundaries of the control parameters. Moreover, the necessary and sufficient condition that the three-delay controller can stabilize the discrete system is presented.Based on this, the stabilizing region of the three-delay controller is obtained.Considering the actual gene regulatory system, a regulating gene with delayed expression of the mRNAs and proteins is added as a time-delay controller. Noted that transcriptional delays can be increased by inserting junk section into the regulating gene, and translational delays can be by slowing down the protein folding. Then, the practicability of time-delay controllers is verified in this actual system.