非線性系統之分叉特性分析與控制

Abstract

多數的工程系統在工作範圍內均有其性能的極限，超過此極限則系統將無法發揮其正常的功能，甚至於接近此工作極限附近，系統仍可能因干擾而造成嚴重的破壞；此種情況通常是因為系統的非線性特性所造成的，且無法藉由系統的線性化模式分析預測而知。所以為了維持系統可接受的安全邊界，在工程系統的設計中，相關的非線性控制乃成為相當重要的研究課題。本論文主要應用分叉（bifurcation）理論與李普諾夫（Lyapunov）理論分析在實際工程系統中非線性現象的穩定性，並提出其相關的控制法則。首先，針對飛機的縱向飛行動態研究，我們成功地應用穩定性判斷法則與分叉理論分析系統的穩定性與可能發生的非線性現象。從理論的解析性結果，獲得檢驗系統平衡點與分叉現象的存在性的方法，並提出系統穩定化與分叉現象的控制法則。此外，藉由雙參數的分叉研究，我們可以將縱向飛行動態區分成不同的操控區間，以增進對於縱向飛行動態更深入的瞭解。在壓縮機系統之應用研究方面，針對未知的特性曲線，我們提出一李普諾夫函數以分析系統特性及估測干擾忍受範圍（attraction region）。除此之外，針對離心式壓縮機系統，在具有系統不確定性的因素下，提出一強健控制補償法則以改善系統因極限圈（limit cycle）的發生所造成之振盪現象。最後，在電力電子應用的研究，我們主要建立buck型直流電壓轉換器之離散時間系統模式，並分析buck型直流電壓轉換器的穩定性與可能出現的非線性現象。由理論分析得知，隨著系統參數如電感、電阻或輸入電壓值的改變，系統響應可能存在倍週期分叉現象，且在倍週期分叉串連發生之後，系統可能進而產生渾沌（chaos）現象。藉由離散時間控制器的設計可以消除或改善由渾沌現象所造成的不穩定的系統響應以提昇系統性能。本論文對於直流電壓轉換器的非線性現象的分析將有助於提昇未來高性能電壓轉換器的設計能力。

Engineering systems are known to have performance limits on their operating beyond or close to these limits, which the system may undergo destructive events even though it operates within the corresponding physical limits. It is known that such phenomena are generally caused by system's nonlinear effect, which cannot be predicted from system linearization. In order to maintain an acceptable safety margin, the nonlinear control of these systems becomes an important and challenge issue. This dissertation applies the bifurcation theory and Lyapunov theory to study the stability analysis and control of nonlinear phenomena in engineering systems. First, the stability and nonlinear behavior of longitudinal flight dynamics is analyzed by applying Routh-Hurwitz stability criteria and bifurcation theorems to the third-order model of longitudinal flight dynamics. The analytic results are obtained for checking the existence of system equilibria and local bifurcations such as Hopf bifurcation and saddle-node bifurcation with respect to the variation of the elevator deflection angle. The control laws using washout filter design are proposed to stabilize system equilibria or to guarantee the stability of periodic solutions emerged from Hopf bifurcation points. The two-parameter bifurcation analysis divides the longitudinal flight space into several maneuvering regions, which might provide more understanding of longitudinal flight dynamics. In the study of axial flow compressor, a Lyapunov function is constructed to analyze system stability and estimate the domain of attraction for compressors with different axisymmetric characteristics. The global stabilization of a centrifugal compressor with spool dynamics is also presented. This is attained by the design of various control schemes for systems with and without system uncertainties. Finally, the sampled-data approach is applied to the study of nonlinear phenomena in buck converters. The period-doubling bifurcations are found to exist as system parameter varies. The bifurcation diagrams for different parameters are also presented to get the insight of converter's nonlinear behavior. Moreover, a series of period-doubling bifurcations might occur which leads to a step-wise transition from period-one to chaos. The bifurcation control using discrete-time washout filter is applied to remove or shift the bifurcation point. From the understanding of chaotic behavior, new possibilities of operating regimes are opened up for the optimal design of converters.