Dengeli gerçekleme ile model indirgeme
Dengeli gerçekleme ile model indirgeme
Dosyalar
Tarih
1996
Yazarlar
Özaslan, Emir
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Özet
Bu tezde, yüksek dereceli modellenmiş sistemlerde dengeli gerçekleme kesme yöntemi ile derece indirgeme ele alınmış, yaklaşıklığı hatasının H*, ve H2 normları incelenmiştir. Model indirgemede atılacak durumların seçilmesinde dikkate alınan kriterler belirlenip, dengeli gerçeklemenin bu kriterler açısından önemi açıklanmıştır. Dengeli gerçekleme kesme yöntemi ile model indirgeme yöntemi gösterilmiştir. Minimum gerçeklenmiş durum uzayı modeli verilen bir sistemi dengeli gerçekleyen koordinat dönüşüm matrisinin bulunması için farklı yöntemler verilmiştir. Model indirgemede indirgenmiş sistemin gerçek sistemi ne derece yaklaşıklıkla temsil ettiğini belirlemek için iki sistem arasındaki hatanın H«, ve H2 normları incelenmiştir. Hatanın H» normu için üst ve alt sınır verilip, hatanın H2 normunun küçültülmesi için bir yöntem önerilmiştir. Model indirgemenin uygulama alanlarından filtre tasarımında, yüksek dereceli FIR filtrelerin, dengeli gerçekleme kesme yöntemi ile daha düşük dereceli IIR filtrelere dönüştürülmesi gösterilmiştir.
In many engineering applications, the choice of a model for a physical process of interest is often complicated by competing desirable goals. On the one hand one would like a model which reflects the characteristics and predicts the behavior of the actual physical device with a useful amount of accuracy. On the other hand one would like the model to be sufficiency simple so as to make various numerical procedures and computational algorithms practical and easy. Unfortunately making to model more realistic and accurate usually leads to increasing the number of parameters needed to describe it which in turn leads to greater and greater complexity for computational manipulation. If some of the parameters in fact have practically very little influence on the model, it takes good physical sense to discard them. More generally, the problem of model reduction is concerned with approximating ( in some appropriate sense ) a given model by a model which is less complex and thereby easier to work with. There are many application areas of model reduction in engineering systems. For example (i) Plant models can be derived by combining models of the separate components, hence forming a single model of high order. If a reduced order model could be substituted then a subsequent control system design would be computationally less demanding and possibly numerically more reliable. (ii) If a high order dynamic compensator has been designed for a plant then a reduced order model of the compensator would involve fever components or computing resources in an implementation. (iii) In filter design it is sometimes possible to simply produce satisfactory high order filters and reduced order filters would then save in implementation.(iv) In modeling some distributed parameter systems high order approximate models can be produced from finite element analysis or modal analysis. Many model reduction methods for linear dynamical systems have been proposed. They have mostly been based on modal analysis or frequency domain concepts. Recently, a method based on measures of controllability and observability has been suggested. In this method, the most controllable and observable part is used as a low order approximation for the model. The controllability and observability grammian as used to define measure of controllability and observability in certain directions of the state space. In the state space model of a system, the minimum input energy needed to control the states is given below. J(Uopt) = x0*p-1x0 and the output energy that indicate the effects of the state on output is given below 1/(0X0* = *o*0*o The controllability P, and the observability Q gramians aren't invariant under coordinate transformations and it is shown that there exists a coordinate system in which the grammians are equal and diagonal as follows P = Q = diag(ava2,...,cxn), oj > ct/+1 > 0 Hence, the corresponding system, representation is called balanced. In the balanced system, the value of the diagonal entries of gramians determines the minimum input energy needed to control the corresponding states and the effects of the corresponding states on the output energy. Since the states corresponding to the smallest a-, entries, are the least controllable and at the same time, the least observable reduced order model is obtained by deleting these states. This model reduction method is called " Balance and Truncate " method. The gramians of the full order balanced system that is asymptotically stable have a form as follows p=e= S, oIf Zı and E2 have no diagonal entries in common then the reduced order system obtained by using balance and truncate method, is asymptotically stable, controllable and observable. Two methods are given to find the coordinate transformation matrix used to obtain balanced representation of a system. In the one of the them, it is shown that to transform the matrix PQ diagonal form is sufficient in the model reduction therefore it isn't necessary to find balanced representation of a system. Here, P and Q are respectively the controllability and observability matrix of a system. In this new proposed method, the similarity transformation matrix that make the PQ matrix diagonal is firstly found. When the coordinate transformation is applied to the system by using the similarity transformation matrix, P and Q gramians are also transformed the diagonal form. Hence, the system obtained from the result of this first coordinate transformation using the similarity matrix can be reduced the lower order by deleting the states corresponding to the smallest diagonal entries of PQ matrix. It is shown that there is a coordinate transformation between the reduced order system obtained by using the above method, the balanced representation of the system can also be obtained by using the second transformation matrix. Furthermore, it is shown that two balanced realization of a given system are connected with a similarity transformation T which is characterized by the equation below TZ=ET and T*T=I where £ is the balanced gramian of both balanced realization. The Ho, norm and the H2 norm of error between reduced order system and full order system are considered in order to measure the approximation error. The H"o norm of the error indicates the maximum magnitude of error over all frequencies.
In many engineering applications, the choice of a model for a physical process of interest is often complicated by competing desirable goals. On the one hand one would like a model which reflects the characteristics and predicts the behavior of the actual physical device with a useful amount of accuracy. On the other hand one would like the model to be sufficiency simple so as to make various numerical procedures and computational algorithms practical and easy. Unfortunately making to model more realistic and accurate usually leads to increasing the number of parameters needed to describe it which in turn leads to greater and greater complexity for computational manipulation. If some of the parameters in fact have practically very little influence on the model, it takes good physical sense to discard them. More generally, the problem of model reduction is concerned with approximating ( in some appropriate sense ) a given model by a model which is less complex and thereby easier to work with. There are many application areas of model reduction in engineering systems. For example (i) Plant models can be derived by combining models of the separate components, hence forming a single model of high order. If a reduced order model could be substituted then a subsequent control system design would be computationally less demanding and possibly numerically more reliable. (ii) If a high order dynamic compensator has been designed for a plant then a reduced order model of the compensator would involve fever components or computing resources in an implementation. (iii) In filter design it is sometimes possible to simply produce satisfactory high order filters and reduced order filters would then save in implementation.(iv) In modeling some distributed parameter systems high order approximate models can be produced from finite element analysis or modal analysis. Many model reduction methods for linear dynamical systems have been proposed. They have mostly been based on modal analysis or frequency domain concepts. Recently, a method based on measures of controllability and observability has been suggested. In this method, the most controllable and observable part is used as a low order approximation for the model. The controllability and observability grammian as used to define measure of controllability and observability in certain directions of the state space. In the state space model of a system, the minimum input energy needed to control the states is given below. J(Uopt) = x0*p-1x0 and the output energy that indicate the effects of the state on output is given below 1/(0X0* = *o*0*o The controllability P, and the observability Q gramians aren't invariant under coordinate transformations and it is shown that there exists a coordinate system in which the grammians are equal and diagonal as follows P = Q = diag(ava2,...,cxn), oj > ct/+1 > 0 Hence, the corresponding system, representation is called balanced. In the balanced system, the value of the diagonal entries of gramians determines the minimum input energy needed to control the corresponding states and the effects of the corresponding states on the output energy. Since the states corresponding to the smallest a-, entries, are the least controllable and at the same time, the least observable reduced order model is obtained by deleting these states. This model reduction method is called " Balance and Truncate " method. The gramians of the full order balanced system that is asymptotically stable have a form as follows p=e= S, oIf Zı and E2 have no diagonal entries in common then the reduced order system obtained by using balance and truncate method, is asymptotically stable, controllable and observable. Two methods are given to find the coordinate transformation matrix used to obtain balanced representation of a system. In the one of the them, it is shown that to transform the matrix PQ diagonal form is sufficient in the model reduction therefore it isn't necessary to find balanced representation of a system. Here, P and Q are respectively the controllability and observability matrix of a system. In this new proposed method, the similarity transformation matrix that make the PQ matrix diagonal is firstly found. When the coordinate transformation is applied to the system by using the similarity transformation matrix, P and Q gramians are also transformed the diagonal form. Hence, the system obtained from the result of this first coordinate transformation using the similarity matrix can be reduced the lower order by deleting the states corresponding to the smallest diagonal entries of PQ matrix. It is shown that there is a coordinate transformation between the reduced order system obtained by using the above method, the balanced representation of the system can also be obtained by using the second transformation matrix. Furthermore, it is shown that two balanced realization of a given system are connected with a similarity transformation T which is characterized by the equation below TZ=ET and T*T=I where £ is the balanced gramian of both balanced realization. The Ho, norm and the H2 norm of error between reduced order system and full order system are considered in order to measure the approximation error. The H"o norm of the error indicates the maximum magnitude of error over all frequencies.
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996
Anahtar kelimeler
Model indirgeme yöntemleri,
Model reduction methods