Effects of Weighting Matrices on SDRE Based Control Method of 3-DoF Helicopter

Abstract

State Dependent Riccati Equation (SDRE) technique enables a suboptimal control law for a class of nonlinear systems such that it minimizes a given quadratic cost function. A nonlinear system is treated as a linear system by being computed its nonlinear matrices at each instant of time and the optimal control problem of interest can be defined as a Linear Quadratic Regulator (LQR) problem in each instant. Therefore, the weighting matrices of LQR play an important role in shaping the transient time response of the closed-loop system by means of SDRE controller. In this study, a SDRE based optimal controller was designed for controlling the position of a 3 DOF laboratory helicopter. The experiments were repeated with different weighting matrices to evaluate their effects on the transient time response of the helicopter. In the first phase of the experiments, the weighting matrices were selected such that form diagonal matrix with constant real elements. The diagonal elements corresponding to the states of the helicopter controlled by SDRE method were changed to explore how affect the transient time responses of these states. In the second phase, the weighting matrices were selected to be state-dependent. The comparison of the experimental results in both phases reveal that the state dependent weighting matrices have more capabilities of enhancing transient time response specifications such as settling time and steady-state error.

Keywords

• Nonlinear control method
• SDRE control
• Suboptimal control

AĞIRLIK MATRİSLERİNİN 3-SD HELİKOPTERİN DDRD TABANLI KONTROL METODU ÜZERİNE ETKİLERİ

Öz

Durum Değişkenine Bağlı Riccati Denklemi (DDRD) tekniği, verilen ikinci dereceden bir maliyet fonksiyonunu en aza indirecek şekilde doğrusal olmayan bir sistem sınıfı için optimale yakın bir kontrol kanunu sağlar. Doğrusal olmayan sistem (DOS) matrisleri her zaman anında hesaplanıp, DOS doğrusal ve zamanla değişmeyen bir sistem olarak ele alınabilir ve ilgili optimal kontrol problemi her anda Doğrusal Kuadratik Regülatör (DKR) problemi olarak tanımlanabilir. Bu nedenle, DKR'nin ağırlık matrisleri, DDRD denetleyicisi vasıtasıyla kapalı çevrim sistemin geçici zaman cevabını şekillendirmede önemli bir rol oynamaktadır. Bu çalışmada, üç serbestlik dereceli (3-SD) deney helikopterinin pozisyon kontrolü için DDRD tabanlı bir optimal kontrolcü tasarlandı. Deneyler, helikopterin geçici zaman cevabı üzerindeki etkilerini değerlendirilmek için farklı ağırlık matrisleriyle tekrarlandı. Deneylerin ilk aşamasında, ağırlık matrisleri sabit gerçek elemanlı köşegen matris olarak seçildi. DDRD metoduyla kontrol edilen helikopterin durumlarıyla ilişkili köşegen elemanlar, bu durumların geçici zaman cevaplarını nasıl etkilediğini incelemek için değiştirildi. İkinci aşamada, ağırlık matrisleri durum bağımlı olarak seçildi. Her iki aşamadaki deneysel sonuçların kıyaslaması, durum bağımlı ağırlık matrislerinin yerleşme zamanı ve kalıcı durum hatası gibi geçici zaman cevabının özelliklerini iyileştirme yeteneğine daha fazla sahip olduklarını ortaya çıkartmaktadır.

Anahtar Kelimeler

• Doğrusal olmayan kontrol
• DDRD kontrolü
• Optimale yakın kontrol

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