Please use this identifier to cite or link to this item: http://hdl.handle.net/11527/16491
Title: Çok Girişli Sistemlerde Sınırlandırmalı Sonlu Zaman Kontrolu
Other Titles: Constrained Deadbeat Control For Multi-input Systems
Authors: Bir, Atilla
Güzelkaya, Müjde
14156
Kontrol ve Otomasyon Mühendisliği
Control and Automation Engineering
Keywords: Sonlu zaman denetimi
Çok girişli sistemler
Finite time control
Multi-input systems
Issue Date: 1990
Publisher: Fen Bilimleri Enstitüsü
Institute of Science and Technology
Abstract: Sonlu zaman kontrol problemi, belirli bir başlangıç anında, belirli bir başlangıç durumunda bulunan bir sistemi, istenilen bir varış durumuna en küçük adım sayısında iletecek giriş değer dizisinin belirlenmesi problemi olarak tanımlanır. Giriş değer dizisinin belirlenebilmesi sistemin kontroledi lebi I ir olmasını gerektirir. Giriş ve durum değişkenleri sınırlandırılmamış tek girişli ve kontroledi lebi I ir sistemlerde sonlu zaman kontrol probleminin çözümü tekdir. Ancak çok girişli sistemlerde, problemin çok sayıda çözümü ile karşılaşılabilir. Giriş ve durum değişken lerinin sınırlandırılması halinde ise, sistemi varış durumuna ulaştırmak için gerekli adım sayısı artabileceğinden, tek girişli sistemlerde bile, sonsuz sayıda çözüm bulunabilir. Tezde ilk olarak, sonlu zaman kontrolü, giriş ve durum sınırlandırmaları göz önünde bulundurulmadan tanıtılmış ve çok girişli sistemler için problemin çözümüne ilişkin koşullar belirlenerek, sonlu zaman kontrollü sistemlerin zamanda. ayr ık lineer regülatör özellikleri tartışılmıştır. İkinci aşamada, giriş ve durum sınır I andırma I ar inin bulunması halinde, problemin geometrik yorumu yapılmış ve bu sınırlandırmalar sırasıyla değerlendirilerek, çok girişli sistemler için. sıfırdan farklı varış durumlarına da erişilmesini sağlayan bir sonlu zaman kontrol algoritması türe tilmiştir. Ayrıca, sınırlandırılmış girişler için sonlu zaman kontrolü i le minimum zaman kontrolü arasındaki ilişkiye değinilerek, iki probleme ait çözümlerin özdeşleştiği durumlar tartışılmıştır. Türetilen kontrol algoritması, çalışmanın son bölümünde, gerçeğe uygun bir model kurabilmek ve ölçüm yapabilmek amacıyla, analog ve sayısal bilgisayarlar hibrit çalıştırılarak, çeşitli sistemlerin sonlu zaman kontrol l arı na başarıyla uygulanmış ve elde edilen sonuçlar irdelenmiştir.
Deadbeat control can be defined as the determination of a control sequence, so that the control process can be taken from the initial state xo to a desired state xt in a minimum number of time steps. As it is obvious from this description, deadbeat control problem can also be represented as the discrete time optimal control problem. Early contributions on the dead-beat concept, depending on the input-output relations of the systems were given by Ragazzini & Franklin (1958). When the input-output relations were of concern, the problem is defined as the determination of a discrete controller that would force the system output to reach to the system input in a finite time and follow it without any steady-state error. The state space approach to the topic was given by Kalman (1960) who solved the problem of transfering the state of a single input samp led- da t a system from its initial state to the origin in a minimum number of time steps using a linear state feedback. So, the problem was treated as the discrete time optimal control problem. In solving this problem, Kalman has also introduced the concepts of controllability and observability. Then, many authors contibuted to the subject and various studies such as deadbeat control of mul t ivar iable systems, input and state constrained deadbeat control, inaccesible state deadbeat control, deadbeat control of time-varying systems deadbeat control in non-linear systems are still under research. This study deals with input and state constrained deadbeat control problem and is divided into 4 chapters as the following: Chapter 1 : Introduction Chapter 2 : Representation of The Deadbeat Control Problem and Its Properties Chapter 3 : Input and State Constrained Deadbeat Control Problem Chapter 4 : Modelling of The Constrained Deadbeat Controlled on a Hybrid Computer In the first chapter, studies concerning the problem are sumnar i zed and some fundamental notions about the problem are given. Chapter 2 is devoted to the representation of the deadbeat concept and its properties. In this chapter, unconstrained deadbeat control problem is investigated and its fundementals are given in Section 2.1. Based on the controllability criterion, a controllable system can be taken from a given initial state to a desired final vi i i state in u time steps, where u. is the contol labi I i ty index of the system. When the input and state variables are unconstrained, u defines the least number of time steps, required for the determination of the input sequence that will force the system to a desjred final state. If a system that is taken from an initial state xo^x^ko) to a final state xt=x>(kt) in u time steps is considered, it is obvious that, the minimum number of the time steps required to reach from any state x(k) (ko<k where [ZM] is the control labi I i ty matr ix and the vector XM is composed of the initial and the final states. For single input systems, the contol labi I i ty index is equal to the order of the system n, and the deadbeat control can be realized in n time steps. As it is mentioned in Section 2.2, when multi-input systems are of concern, the problem is some more complicated, since the solution deals with the selection of n linearly independent columns of the contol labi I i ty matrix. There are many selection procedures and the deadbeat solutions of the mul t i-input systems are not uniqe. The linearly independent columns of the controllability matrix can be selected by defining various transformation matrices. The order of the transformat ion matr ix and values of its elements define the selection procedure. It is usually desired for the system to stay at the final position after it has reached to it. For this purpose, the final state has to be an equilibrium state of the system to be control led. Transfer ing the states of a system from any initial state to the origin in a minimum number of time steps using a linear state feedback can be formulated as a discrete linear regulator problem. Deadbeat control in mul t i-input systems as a linear state feedback is investigated in Section 2.3. Since the solution to the multi-input case is not unique, various forms of state feedback controllers can be found in literature. It is necessary to use an ordered selection of the linearly independent columns of the control labi I i ty matr ix in order to obtain a state feedback gain matrix and the control law is given by u(k) - [F] x(k) where [F] is the feedback gain matrix. The closed loop system matrix [Qk] thus obtained has the following properties: 1- [Qui is a ni Ipotent matrix of which the index of nilpotency is the system ix nilpotency index u, 2- The eigen values of [Qk] are zero, 3- The matrix [Qk] is similar to the Jordan matrix, composed of r nilpotent Jordan bloks where r is the number of inputs. In practice, because of the physical constraints, the elements of the input sequence and the the states of systems can not take values above some saturation limits. For this reason, the input sequences for the deadbeat control has to be calculated taking these limits into consideration. Chapter 3 is devoted to the input and state constrained deadbeat control problem. When the input and state variables are constrained, the desired forcing effort applied to the process is restricted by the saturation limits and the minimum number of time steps required to take the process from xo to xt would be more than u which is valid for the unconstrained case. Input and state constrained deadbeat control problem can be defined as the determination of a control sequence, so that the control process can be taken from the initial state xo to a desired final state xt in a minimum number of time steps and, for all the elements of the input sequence and the system states under the following constraining conditions Umin</k
Description: Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990
Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1990
URI: http://hdl.handle.net/11527/16491
Appears in Collections:Kontrol ve Otomasyon Mühendisliği Lisansüstü Programı - Doktora

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