Lineer neutral bir sistem için durum geri beslemeli kontrol problemi

Özet

Bu çalışmada durumunun gecikmesi zaman değişkenli, durumunun zamana göre birinci türevinin gecikmesi sabit olan lineer neutral bir sistemin asimptotik kararlılığı için yeterli koşullar Lyapu-nov-Krasovskii yaklaşımı ile sistemin durumundaki gecikmeye bağımlı, durumunun zamana göre birinci türevindeki gecikmeden bağımsız olarak Lineer Matris Eşitsizliği (LME) cinsinden elde edilmiştir. Daha sonra sistemi asimptotik kararlı hale getirmek için, durum geri besleme kontrol kuralı bir kontrol girdisi vasıtası ile sisteme uygulanarak, kapalı çevrim sistemi elde edilmiştir. El-de edilen sistemin asimptotik kararlılığı için yeterli koşullar yine Lyapunov-Krasovskii yaklaşımı ile sistemin durumundaki gecikmeye bağımlı, durumunun zamana göre birinci türevindeki gecikmeden bağımsız olarak LME cinsinden elde edilmiştir. Elde edilen LME, Matlab LMI Toolbox paket prog-ramı ile çözülmüştür. Bu çalışmada, literatürdeki benzer çalışmalarda bulunan durum gecikmesinin zamana göre birinci türevinin birden küçük bir sayıya eşit olma şartı ortadan kaldırılmıştır. Burada yeterli koşullar simetrik bir LME’nin negatif belirli bir matris olması ile elde edilmiştir. Elde edilen LME’nin negatif belirli olması için köşegenindeki tüm elemanlarının negatif olmaları gerekmekte-dir. Durum gecikmesinin zamana göre birinci türevi birden büyük bir sayı olduğunda LME’nin kö-şegenine negatif olmayan bir terim gelmektedir. Burada bu sorunu ortadan kaldırmak için LME’yi elde etme aşamasında elde edilen eşitsizliğe sıfıra eşit olan bir terim eklenerek LME’nin sorun çı-karan köşegen elemanına serbest olarak seçilebilen bir terimin eklenmesi sağlanmıştır. Son olarak elde edilen sonuçlar örnekler üzerinde uygulanıp elde edilen sayısal değerler diğer çalışmalarla karşılaştırılmıştır.

Abstract

The mathematical system representations of some physical and biologic control problems depend also on the states of the systems in the past. Systems of this type are called as time delay systems. The delay may also be on the first time derivative of the states of the systems. These types of systems constitute the general form of the time delay systems and are named as neutral systems. The delay, depending on the system considered can be constant or time vary-ing, and the time delay systems are investigated as a separate class of dynamical systems. The existence of the time delay in any system, can cause to insta-bility and bad performance. This is the reason why the stability and the performance analyses of this type of systems became important both in the theo-retical and the practical means. The time delay sys-tems are separated to two classes as, systems that can be represented with, a system of first order ODE’s and the system of first order PDE’s. In this article the time delay systems belonging to the first class are investigated.. In the years 1990’s, the time delay systems are clas-sified as independent of the delay and as depended to the delay. In the case of independent of the delay, the system is stable for all positive values of the de-lay but for dependent to delay systems, the system is stable for only some positive values of the delay and unstable for the other values. In this article, at first, the model representation of a linear time delay system is introduced, in which the state has a time varying delay and the first time de-rivative of the state has a constant delay. The suffi-cient conditions for the asymptotic stability of the system is obtained as linear matrix inequality by the Lyapunov-Krosovskii approach as dependent to the delay of the state of the systems but independent of the delay in the first time derivative of the state. Ac-cording to the Lyapunov-Krasovskii approach, the system is stable if the first time derivative of the Lyapunov-Krasovskii functional which is construct-ed for the system is less than zero. In this work, by writing the first time derivative of the functional in terms of a vector and a matrix, it is shown that it is less than zero by obtaining the matrix as a nega-tive defnite matrix. Aiming to stabilize the time delay systems, state feedback control problems are investigated in the literature. The state feedback control problem is to investigate the closed loop system which can be ob-tained by appyling a control input to the system. In state feedback control problems the control input is constructed with the state of the system and a regu-lating matrix. In this article, it is also constructed a state feedback control problem for the system given in the first sec-tion. The sufficient conditions for the asymptotic stability of the closed loop system is abtained in terms of linear matrix inequality again by the Lya-punov Krasovskii approach as dependent to the de-lay of the state of the system but independent of the delay in the first time derivative of the state. In the literature,in most of the articles, there is a restriction on the first time derivative of the delays of time delay systems as it has to be less than one. In this article, this restriction is removed by adding a term which is equal to zero to the first time deriva-tive of the Lyapunov-Krasovskii functional. Here the sufficient conditions are obtained with the matrices which are negative definite symetric matrices. For these types of matrices it is necesarry that their ele-ments on the diagonal are all negative. Here, when the first time derivative of the delay is greater than one, nonnegative terms appear on the diagonal of the matrices. This problem is solved by adding a ze-ro term on the first time derivative of the Lyapunov-Krasovskii functional. This provides to define a free parameter on the diagonals in order to make them negative in spite of a first time derivative of the de-lay function that is greater one. At last, the theoretical results obtained are applied to examples and the values obtained for the upper limit of the time delay are given in tables.