Discrete Time State Estimation with Kalman Filter and Adaptive LQR Control of a Time Varying Linear System

Abstract

In this study, a new adaptive controller design was created that compensates for variable load effects and provides high control performance. In the proposed control method, Discrete Time Kalman Filter method (DKF), which estimates system output states, and Discrete Time Linear Quadratic Regulator (DLQR) method, one of the optimal control methods, were used. Although the DLQR control method produces good results when applied to unvarying systems, it cannot provide the desired response in time varying systems because it has no adaptation mechanism. In order to solve this problem, an adaptation mechanism based lyapunov method which has been developed that adapts to different environmental conditions, constantly updating a new state feedback gain matrix value (Knew) and optimal lyapunov adaptation gain values ( theta1, theta2, theta3, theta4, theta5 and theta6) used for system control block such as position (x1) control, speed (x2) control and current (x3) control. In this mechanism, lyapunov adaptation gain initial values were calculated using the Artificial Neural Network (ANN) method as a new control approach. Thus, it was aimed to minmize the effects of variable load and to increase the control system stability. In order to show the proposed method effectiveness, a variable loaded VsimLabs (Virtual Simulation laboratories) servo system was modelled as a time varying linear system and applied in practical implementation and simulation in environment of Matlab Simulink. Taking reference the results of experimental platform and measurement of performance index such as Integral Square Error (ISE), Integral Absolute Error (IAE) and Integral time absolute error (ITAE), it was observed that the proposed control method increases the system stability and performance by eliminating variable load effect and steady state error.

Keywords

• Adaptation mechanism
• Artificial neural network
• Lyapunov method
• Time varying linear system

Kalman Filtresi ile Ayrık Zamanlı Durum Tahmini ve Zamanla Değişen Doğrusal Bir Sistemin Adaptif LQR Kontrolü

Öz

Bu çalışmada, değişken yük etkilerini kompanze eden ve yüksek kontrol performansını sağlayan yeni bir adaptif denetleyici tasarımı gerçekleştirilmiştir. Öne sürülen kontrol metodunda, sistem çıkış durumlarını tahmin eden ayrık zamanlı kalman filtresi (Discrete Time Kalman Filter, DKF) ve optimum kontrol yöntemlerinden biri olan Ayrık Zamanlı Doğrusal Kuadratik Regülator (Discrete Time Linear Quadratic Regulator, DLQR) metodlarından yararlanılmıştır. DLQR kontrol metodu zamanla yükü değişmeyen sistemlere tüm periyotlarda uygulandığında iyi sonuçlar üretmesine rağmen, adaptasyon mekanizması bulunmadığından, zamanla değişen sistemlerde istenilen cevabı verememektedir. Bu problemi çözmek için, farklı çevre ortamlarına uyum sağlayan, yeni bir durum geri besleme kazanç matrix değerini (Knew) ve pozisyon (position, x1) kontrol, hız (speed, x2) kontrol ve akım (current, x3) kontrol gibi sistem kontrol blokları için kullanılan optimum lyapunov adaptasyon kazanç değerlerini ( theta1, theta2, theta3, theta4, theta5 ve theta6) sürekli güncelleyen bir lyapunov tabanlı adaptasyon mekanizması yöntemi geliştirilmiştir. Bu mekanizmada lyapunov adaptasyon kazancın başlangıç değerleri, tasarımda yeni bir yaklaşım olarak Yapay Sinir Ağı (Artificial Neural Network, ANN) metodu ile hesaplanmıştır. Böylece değişken yük etkilerinin minimize edilmesi ve sistem kararlılığının artırılması amaçlanmıştır. Önerilen yöntemin etkinliğini pratik uygulama ve simülasyonda göstermek için, zamanla değişen doğrusal bir sistem olan değişken yüklü bir Sanal Simülasyon laboratuvarları (Virtual Simulation Laboratories, VsimLabs) servo sistemi modellenmiş ve Matlab Simulink ortamında kullanılmıştır. Deneysel sonuçlara ve İntegral Karesel Hata (Integral Square Error, ISE), İntegral Mutlak Hata (Integral Absolute Error, IAE), İntegral Zamanlı Mutlak Hata (Integral time absolute error, ITAE) gibi performans ölçümlerine göre, önerilen yöntemin değişken yük etkisini ve sürekli durum hatasını minimize ederek sistem performans ve kararlılığını artırdığı görülmüştür.

Anahtar Kelimeler

• Adaptasyon Mekanizması
• Yapay Sinir Ağı
• Lyapunov Yöntemi
• Zamanla Değişen Doğrusal Sistem

Kaynakça

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