Stability Analysis Of Several Classes Of Time Delay Systems Based On The ? Decomposition Method

The dynamic systems in the real world can be described by the delay differ-ential equations better, thus the stability analysis of time delay system is always the hot topic in several fields. For the complexity of time delay systems itself, the ultimate knowledge about the stability of such systems is far from achieved even though too many valuable and meaningful works have been done under the effort made by domestic and foreign scholars. Thus, much more energy for systematic analysis is need both in depth and breadth. For the linear time invariant (LTI) delayed system, the stability of the system itself lies in the distribution of systems' characteristic roots. However, delayed system has infinitely many roots for the ex-istence of the exponential term in the characteristic equation. At present, there are two lines to deal with such problem, one is the Nyquist-like algorithm, by deterging the number of roots in the right half complex plane directly, the other is the ?-decomposition method, by dividing the parameter space according to the number of unstable roots, the we can give the stable regions. This dissertation starts from the characteristic quasi-polynomial of the time delay system. The effect of time delay is obtained associating with the ?-decomposition technique. The ?-decomposition method consists of two parts. One is the computation of the boundary at which the stability switches, the other is cross direction over those boundaries. Following these two main topics, some work are complete both in the improvement and the application of the ? decomposition method. And we list them as follows:1. For a class of linear time-invariant delay systems with commensurate delays, the asymptotic behaviors of multiple purely imaginary roots is studied, and the stability interval is given. Due to the appearance of PIRs, the classic tool of singularity analysis in the algebraic geometry, Newton diagram, is introduced to deal with the corresponding problem and the Puiseux series for the multivalued root loci function are given in the critical point. Meanwhile, Weierstrass preparation theorem, a universal reduction result, is also borrowed to simplify the holomorphic function into algebraic function, which make the our analysis much easier.2. The effect of a class of delayed neural networks in a ring topology is con-cerned, and the local stability and HOPF bifurcation are considered at its steady state solution. Firstly, the characteristic quasi-polynomial of the origi-nal system around an equilibrium point has some new properties according to our discussion. Secondly, the exponential polynomial has at most one purely imaginary root which is proofed by the vector addition, and the only PIR can be determined by the graphical test we have proposed. Associating with the initial stable condition, the DNNs are divided into four types, delay dependent stable type, delay dependent unstable type, delay independent stable type, de-lay independent unstable type. At last, as indicated in the HOPF bifurcation theorem, the bifurcation point of HOPF is also given in explicit formula.3. For the first order delayed unstable processes, the relation of time delay and controller parameters is analyzed, and the complete construction for PID con-troller stabilization domain is determined. According to the closed loop char-acteristic equation of the unstable process, the partition of the controller pa-rameter and the delay parameter is completed. At last, all the stabilized region of the PID parameter space is obtained. It is found that, there exists at most one purely imaginary root for this low order system through our discussion. Associating with the initial stability condition, the PID parameter space is divided into four regions as follows:delay dependent stable region, delay de-pendent unstable region, delay independent stable region, delay independent unstable region. In delay dependent stable region, the explicit formula of the maximum allowable time delay is given for the feasible controllers. That is to say, the stabilized region of the PID parameter is completed.4. The ultimate stability of a class of linear system with commensurate delays is considered. First, the characteristic quasipolynominal is reformulated into a factorization form. Then, it is found that 1) the crossing direction of imaginary roots (CDIRs) can be reflected by an appropriate frequency-sweeping test, 2) the CDIRs are determined by a certain order of the imaginary roots by magnitude and finally,3) a time-delay system is ultimately unstable if there is at least one crossing imaginary root. To apply the properties 1)-3), an easily implemented frequency-sweeping method is proposed, by which the CDIRs can be obtained without any further computation. The method can be used for the complete stability of the system. In singular case under certain conditions, some benefit result is the MIR case is also obtained.5. This dissertation presents an effective numerical procedure to analysis the bounded input bounded output (BIBO) stability of a class of commensurate delayed fractional order systems with rational order, and stability intervals of time delay is computed. As is indicated that, the BIBO stability can be represented by the distribution of the system's characteristic roots. Thus, the practical frequency sweeping method can be applied into the computation on stability interval of such fractional order delayed system. Under our discussion, the effectiveness and correctness is proofed and some benefit properties are also obtained. In sum up, the stability interval of time delay is determined completely.