İşaret-akış grafının kazancının ve bir parametreye göre duyarlığın bilgisayarla sembolik olarak bulunması
Symbolic evaluation of signal-flow graph gain and gain sensitivity according to parameter
- Tez No: 39620
- Danışmanlar: PROF.DR. CEVDET ACAR
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Duyarlılık analizi, Grafikler, İşaret akış grafı, Sensitivity analysis, Graphics, Signal flow graph
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu tezde, en genel halde sembolik dal kazançlarına sahip işaret-akış graflanna ait graf kazançlarının ve bu kazançların grafta sadece bir dal kazancında bulunabilen herhangi bir parametreye göre duyarlığının sembolik olarak hesaplanması konusu ele alınarak, bu amaçla sembolik işlemler yapabilen bir program yazılmıştır. İkinci bölümde, işaret-akış grafinın kazana ve bu kazancın graftaki bir parametreye göre duyarlığının hesaplanması konulan ele alınmıştır. Önce işaret-akış grafi kavramı tanıtılmış ve işaret akış grafina ait çeşitli tanımlamalar verilmiştir. Duyarlık hesaplanırken Lynch 'in Duyarlık Formülü 'nden yararlanılmıştır. Üçüncü bölümde, sembolik kazanç ve duyarlık hesaplamaları yapan SFG programının yazılmasında izlenen algoritma incelenmiş ve program tanıtılmıştır. Bu algoritmaya göre, önce işaret-akış grafindan elde edilebilecek tüm çevreler bulunmuştur. Bu çevreler yardımıyla değmeyen çevre grupları oluşturularak işaret-akış grafinın determinantı sembolik olarak elde edilmiştir. Ardından kapalı bir işaret-akış grafi oluşturulmuş ve bu graftan yararlanılarak kazancın pay ifadesi elde edilmiştir. Kazancın bir parametreye göre duyarlığının hesaplanabilmesi için önce Lynch 'in Duyarlık Formülü kullanılarak, kazancın parametrenin bulunduğu dal kazancına göre duyarlığı hesaplanmıştır. Ardından elde edilen bu sonuç, dal kazancının parametreye göre duyarlığı ile çarpılarak istenilen sonuç sembolik olarak elde edilmiştir.
Özet (Çeviri)
The linear network model includes a number of components : resistors, capacitors, transformers, gyrator, etc. Each component of the network model has been characterized by a parameter and by a graphical symbol. Networks are made up of interconnections of these components. Each voltage-current relationship defines a mathematical operation. An operational symbol could be used to represent the mathematical operation, and these operational symbols could be interconnected into an operational diagram. The analysis of this operational diagram would provide an alternative means for determining transfer functions of networks. A signal flow graph is just such an operational diagram which represents a system of simultaneous linear equations: yf = Z tij Yi, i = l,2,3,...,m<n in which Yj represents i th variable in the equations, tij represents the coefficient of the i th variable in the j th equation. A signal-flow graph consists a number of nodes, which represent the variables, connected by directed branches along which the signal flows in the direction of arrow. Node variables represent variables. Branch transmittances represent coefficients in the relationships among variables. Each node adds algebraically all the signals directed toward the node and transmits this sum along each branch directed away from node. Thus the variable represented by a node is the sum of all signals entering the node. The signal leaving a node by a branch is multiplied by the transmittance of that branch in its passage along the branch. In a signal-flow graph, a node with only outgoing branches is called source node. A node with some incoming branches is called a VIdependent node. In particular, a dependent node with only incoming branches is called a sink node. Source node variables represent independent variables in the equations. Dependent node variables represent dependent variables in the equations. If all the source node variables set equal to zero except yh then any dependent node variable y“j is expressed as follows, Yj = Tij. y; in which Ty is called the transmission or transfer function from the source node variable y, to the dependent node variable y”j. The following is a topological rule for evaluating Tij [1], [2], [3] : ZTkAk TH = -*- where the graph determinant A is given by, A = 1“ £ Pta + S P2m ”- + (-l)k £ Pkm in which Pkm is the product of m th set loop transmittances of the k nontouching loops. Tk is the path transmittance of the k th path from y; to yj. Ak is the subgraph determinant which does not touch the k th path from yj to yj. The rule is called“Mason 's Rule ”or“ Mason 's Gain Formula ”. The graph gain expression is a network function expressing the ratio of a transform to an excitation transform. The same network function should be obtained whether it is calculated with the use of a signal flow graph or from a solution of the network equations. It can be calculated sensitivity due to one of the branch transmittances in the signal-flow graph by using the transfer function of a network,. Each parameter in a vunetwork has an influence on the response. As the parameter is changed ( due to aging, temperature changes, humidity etc. ) the response will change. It is of interest to know by what fraction the response will change when a given parameter changes by a certain fraction. This information is given by what is called the sensitivity. It is defined the sensitivity of a quantity, say the transfer function T, to a parameter x by s(T,x)-sî = £IjI-i£I o x / x T o x In this definition the changes are considered to be differentially small. Sensitivity expression can be written based on signal flow graphs and Mason 's Gain Formula. One of the methods is Lynch 's Sensitivity Formula. According to this formula, the sensitivity due to one of the branch transmittances tij is calculated by the following expression : Ao k X (TkAk)o S(T,x) = ^ - A S TkAk k in which A is the graph determinant and denominator of the transfer function calculated by using Mason 's Gain Formula. ZTkAk is the numerator of the transfer function calculated by using Mason 's Gain Formula. A o is the polynomial found by setting the terms have tij in A be zero. Z(Tk Ak) o is the polynomial found by setting the terms have tij in ZTkAk be zero [4]. vmIf tij is a function of parameter x (tij = f(x)) then it can be calculated S(T,x) as follows [4]. S(T,x) = S(T,tij).S( tij,x) where tij ^x Systematic analysis of signal-flow graph by Mason 's Gain Formula requires an algorithm for enumerating all the paths and loops in a network. Such an algorithm is desirable in cases where the network is large or where the analysis is to be done with the aid of the computer. The method is based on a combinational approach to the problem of detecting paths and loops. The first point to clarify is that a loop is really just a special type of path, therefore, the distinction between loops and paths can be disregarded and the general problem of rinding paths can be considered. A path is a sequence of branches, but more basically it 's a combination of nodes. Therefore, if it were possible to determine all of the combinations of all the branches taken 1 through N at a time, where N is the total number of branches, and then determine all of the possible permutations of these combinations all of the possible ( and impossible ) paths and loops would be included. The trick is to form the combinations and permutations in a manner such that no useless information (invalid combinations for paths ) is generated [5]. After finding all the paths, it can be found all the loops. A loop is a kind of path which has a same node at the beginning of the path and the end of the path. So, it can be generated nontouching loop sets by using these loops. These nontouching loop sets get to determine the graph determinant Mason 's rule requires two sets of rules, one each for the denominator and numerator. From a computer programming of view, it would be very convenient to evaluate both denominator and numerator by a single process. This is made possible through the use of the closed signal-flow graph. Suppose that ifs going to be calculated Tij. It 's added one more branch to the original signal-flow graph. This additional branch has a branch transmittance“1”and is directed from node y} to node y{. Such a derived graph is called“ closed signal-flow graph ”. It's evaluated the IXgraph determinant Ac for this closed signal-flow graph such as original signal-flow graph is. The result may be expressed as [6] : Ac = A - ZTkAk ZTkAk = A - Ac k The result shows that if it's evaluated the determinant of the closed signal-flow graph, then the distinct between A and Ac gives the numerator of the desired Tij.
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