SYSTEMS AND THEIR REPRESENTATION: Basic elements in control systems, Open and closed loop systems, Electrical analogy of Mechanical and thermal systems, Transfer function, Synchros, AC and DC servomotors, Block diagram reduction techniques, Signal flow graphs. Signal Flow Graph is one of the most important topics in Control Systems for GATE 2018. Download these short notes in PDF & learn the concepts of drawing Signal Flow Graph. Also know how to find the gain of the system through Signal Flow Graph Reduction method.

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'Mason graph' redirects here. For other flow graphs, see.A signal-flow graph or signal-flowgraph ( SFG), invented by, but often called a Mason graph after who coined the term, is a specialized, a in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of (also called ), which includes as well that of. This mathematical theory of digraphs exists, of course, quite apart from its applications.SFGs are most commonly used to represent signal flow in a and its controller(s), forming a.

Among their other uses are the representation of signal flow in various electronic networks and amplifiers, state-variable filters and some other types of analog filters. In nearly all literature, a signal-flow graph is associated with a.

Contents.History Wai-Kai Chen wrote: 'The concept of a signal-flow graph was originally worked out by 1942in dealing with analog computers. The greatest credit for the formulation of signal-flow graphs is normally extended to 1953, 1956. He showed how to use the signal-flow graph technique to solve some difficult electronic problems in a relatively simple manner.

The term signal flow graph was used because of its original application to electronic problems and the association with electronic signals and flowcharts of the systems under study.' Lorens wrote: 'Previous to 's work, worked out a number of the properties of what are now known as flow graphs. Unfortunately, the paper originally had a restricted classification and very few people had access to the material.' 'The rules for the evaluation of the graph determinant of a Mason Graph were first given and proven by Shannon 1942 using mathematical induction. His work remained essentially unknown even after Mason published his classical work in 1953. Three years later, Mason 1956 rediscovered the rules and proved them by considering the value of a determinant and how it changes as variables are added to the graph.'

Domain of application Robichaud et al. Identify the domain of application of SFGs as follows: 'All the physical systems analogous to these networks constructed of ideal transformers, active elements and gyrators constitute the domain of application of the techniques developed here. Trent has shown that all the physical systems which satisfy the following conditions fall into this category. The finite lumped system is composed of a number of simple parts, each of which has known dynamical properties which can be defined by equations using two types of scalar variables and parameters of the system. Variables of the first type represent quantities which can be measured, at least conceptually, by attaching an indicating instrument to two connection points of the element.

Variables of the second type characterize quantities which can be measured by connecting a meter in series with the element. Relative velocities and positions, pressure differentials and voltages are typical quantities of the first class, whereas electric currents, forces, rates of heat flow, are variables of the second type. Firestone has been the first to distinguish these two types of variables with the names across variables and through variables. Variables of the first type must obey a mesh law, analogous to Kirchhoff's voltage law, whereas variables of the second type must satisfy an incidence law analogous to Kirchhoff's current law. Physical dimensions of appropriate products of the variables of the two types must be consistent. For the systems in which these conditions are satisfied, it is possible to draw a linear graph isomorphic with the dynamical properties of the system as described by the chosen variables.

The techniques. can be applied directly to these linear graphs as well as to electrical networks, to obtain a signal flow graph of the system.' Basic flow graph concepts The following illustration and its meaning were introduced by Mason to illustrate basic concepts. — Robichaud Non-uniqueness Robichaud et al. Wrote: 'The signal flow graph contains the same information as the equations from which it is derived; but there does not exist a one-to-one correspondence between the graph and the system of equations.

One system will give different graphs according to the order in which the equations are used to define the variable written on the left-hand side.' If all equations relate all dependent variables, then there are n! Possible SFGs to choose from. Linear signal-flow graphs Linear signal-flow graph (SFG) methods only apply to, as studied.

When modeling a system of interest, the first step is often to determine the equations representing the system's operation without assigning causes and effects (this is called acausal modeling). A SFG is then derived from this system of equations.A linear SFG consists of nodes indicated by dots and weighted directional branches indicated by arrows. The nodes are the variables of the and the branch weights are the coefficients. Signals may only traverse a branch in the direction indicated by its arrow. The elements of a SFG can only represent the operations of multiplication by a coefficient and addition, which are sufficient to represent the constrained equations. When a signal traverses a branch in its indicated direction, the signal is multiplied the weight of the branch.

When two or more branches direct into the same node, their outputs are added.For systems described by linear algebraic or differential equations, the signal-flow graph is mathematically equivalent to the system of equations describing the system, and the equations governing the nodes are discovered for each node by summing incoming branches to that node. These incoming branches convey the contributions of the other nodes, expressed as the connected node value multiplied by the weight of the connecting branch, usually a real number or function of some parameter (for example a variable s).For linear active networks, Choma writes: 'By a 'signal flow representation' or 'graph', as it is commonly referred to we mean a diagram that, by displaying the algebraic relationships among relevant branch variables of network, paints an unambiguous picture of the way an applied input signal ‘flows’ from input-to-output.

A motivation for a SFG analysis is described by Chen: 'The analysis of a linear system reduces ultimately to the solution of a system of linear algebraic equations. As an alternative to conventional algebraic methods of solving the system, it is possible to obtain a solution by considering the properties of certain directed graphs associated with the system.' See subsection:. 'The unknowns of the equations correspond to the nodes of the graph, while the linear relations between them appear in the form of directed edges connecting the nodes.The associated directed graphs in many cases can be set up directly by inspection of the physical system without the necessity of first formulating the →associated equations.' Basic components.

— Robichaud, Signal flow graphs and applications, 1962For digitally reducing a flow graph using an algorithm, Robichaud extends the notion of a simple flow graph to a generalized flow graph:Before describing the process of reduction.the correspondence between the graph and a system of linear equations. Must be generalized. The generalized graphs will represent some operational relationships between groups of variables.To each branch of the generalized graph is associated a matrix giving the relationships between the variables represented by the nodes at the extremities of that branch.The elementary transformations defined by Robichaud in his Figure 7.2, p. 184 and the loop reduction permit the elimination of any node j of the graph by the reduction formula:described in Robichaud's Equation 7-1. With the reduction formula, it is always possible to reduce a graph of any order. After reduction the final graph will be a cascade graph in which the variables of the sink nodes are explicitly expressed as functions of the sources. This is the only method for reducing the generalized graph since Mason's rule is obviously inapplicable.

— Robichaud, Signal flow graphs and applications, 1962The definition of an elementary transformation varies from author to author:. Some authors only consider as elementary transformations the summation of parallel-edge gains and the multiplication of series-edge gains, but not the elimination of self-loops.

Other authors consider the elimination of a self-loop as an elementary transformationParallel edges. Replace parallel edges with a single edge having a gain equal to the sum of original gains. Signal flow graph of a circuit containing a two port. The forward path from input to output is shown in a different color. The dotted line rectangle encloses the portion of the SFG that constitutes the two-port.The figure to the right depicts a circuit that contains a. V in is the input of the circuit and V 2 is the output. The two-port equations impose a set of linear constraints between its port voltages and currents.

The terminal equations impose other constraints. All these constraints are represented in the SFG (Signal Flow Graph) below the circuit. There is only one path from input to output which is shown in a different color and has a (voltage) gain of -R Ly 21. There are also three loops: -R iny 11, -R Ly 22, R iny 21R Ly 12. Sometimes a loop indicates intentional feedback but it can also indicate a constraint on the relationship of two variables. For example, the equation that describes a resistor says that the ratio of the voltage across the resistor to the current through the resistor is a constant which is called the resistance. This can be interpreted as the voltage is the input and the current is the output, or the current is the input and the voltage is the output, or merely that the voltage and current have a linear relationship.

Virtually all passive two terminal devices in a circuit will show up in the SFG as a loop.The SFG and the schematic depict the same circuit, but the schematic also suggests the circuit's purpose. Compared to the schematic, the SFG is awkward but it does have the advantage that the input to output gain can be written down by inspection using.