CONTROL ENGINEERING SYSTEM 1

CONTROL ENGINEERING SYSTEM 1

I. Signal Flow Graphs - Basics

An in depth discussion of signal flow graphs and their manipulation can be found in:

Howard R., Dynamic probabilistic systems, John Wiley, New York, 1971.
Truxal, JG., Automatic feedback control system synthesis. McGraw-Hill, New York, 1955.

A. Rules and Definitions of Signal Flow Graphs

Signal flow graphs follow four rules:

Signals travel along branches only in the direction of the arrows.
A signal travelling along any branch is multiplied by the transmission of that branch.
The value of any node variable is the sum of all signals entering the node.
The value of any node variable is transmitted on all branches leaving that node.

A path is a continuous succession of branches, traversed in the indicated branch directions. The path transmission is defined as the product of branch transmissions along the path. A loop is a simple closed path, along which no node is encountered more than once per cycle. The loop transmission is defined as the product of the branch transmissions in the loop.

The transmission T of a flow graph is defined as the signal appearing at some designated dependent node per unit of signal originating at a specified source node. Specifically, T_ik is defined as the signal appearing at node k per unit of external signal injected at node j. There are a number of ways of computing transmissions.

B. Basic Operations on Signal Flow Graphs

Solution of signal flow graphs requires knowledge of certain of their topological properties. The basic operations of addition, multiplication, distribution and factoring can be used to reduce the number of branches and nodes in the system. At first glance, it might appear that by successive application of such transformations a graph could be reduced to a single branch connecting any two given nodes. However, if the graph contains a closed loop of dependencies, as when modeling iterations, one or more self loops will eventually appear.

C. The Effect of a Self Loop

The effect of a self loop at some node on the transmission through that node is analyzed in the figure below.

Effect of a self loop

The node signal at the first node is x and the signal returning around the self loop is xt. Since the node signal is the algebraic sum of the signals entering that node, the external signal arriving from the left must equal y(1-t). Hence, the effect of a self loop t is to divide an external signal by the factor (1-t) as the signal passes through the node. This holds for all t.

D. Solution by Node Absorption

Node absorption corresponds to the elimination of a variable by substitution in the associated algebraic equations. With the aid of the basic transformations and the self loop replacement, any node in a graph can be absorbed and the equivalent expressions for the transmission between two other nodes calculated. Although the branch is no longer shown, its effects is included in the new branch transmission values, as shown below.

Absorption of a node

To compute the overall graph transmission, all the intermediate nodes are absorbed in turn, yielding the transmission between the start and finish nodes. reduction of graphs is computationally intensive and manual solution of graphs of even moderate size can be difficult.

. Modeling Design Iteration Using Signal Flow Graphs
The signal flow graph represents a diagram of relationships among a number of variables. When these relationships are linear, the graph represents a system of simultaneous linear algebraic equations. The signal flow graph, as shown in the figure below, is composed of a network of directed branches which connect at the nodes. A branch jk, beginning at node j and terminating at node k, indicates its direction from j to k by an arrowhead on the branch. Each branch jk has associated with it a quantity known as the branch transmission P_jk .

For our modeling processes, the branches represent the tasks being worked (an activity -on-arc representation). The branch transmissions include the probability and time to execute the task represented by the branch:

P_jk = p_jk

where: p_jk is the probability associated with the branch.

t_jk is the time taken to traverse the branch

z is the transform variable used to connect the physical system (time domain) to the quantities used in the analysis (transform domain). The z transform simplifies the algebra , as it enables us to incorporate the quantities to be multiplied (probabilities) in the coefficient of the expression, and to include the quantities to be added (task times) in the exponent. The resulting system is then analogous to a discrete sampled data system, and the body of literature on this subject can be applied for the analysis thereof.

The path transmission is defined as the product of all branch transmissions along a single path. The graph transmission is the sum of the path transmissions of all the possible paths between two given nodes. The graph transmission is also the resulting expression on an arc connecting the two given nodes when all of the other nodes have been absorbed. In particular, we are interested in computing the graph transmission from the start to the finish nodes. Henceforth, graph transmission shall refer to the graph transmission between the start and the finish nodes, and is denoted as T_sf.

The coefficient of each term in the graph transmission is the probability associated with the path(s) it represents, and the exponent of z is the duration associated with the path(s). The graph transmission can be derived using the standard operations for the signal flow graphs. The impulse response of the graph transmission is then a function representing the probability distribution of the lead time of the process. It can be shown that the expected value of the lead time of the process is:

Numerical Example

A simple example is shown below:

The hypothetical design process is represented by the graph shown below:

The two tasks A and B (product design and tooling design) take 3 and 2 units of time respectively. Once task B is attempted, task A is reworked with probability 0.6, and once A is attempted, B is reworked with probability 0.3. Iterative repetitions of A are represented by task A'.

The graph transmission is found using the node elimination techniques. This graph transmission is given by:

The expected value of the project lead time E(L) is 7.6 units of time.

The first few terms of the probability distribution function is represented graphically in the figure below:

The distribution can be found for this simple example by performing synthetic division on T_sf to obtain the first few terms of the infinite series. The nominal (once through) time for A and B in series is 5 units of time, which occurs with probability 0.4. It is more likely (probability 0.42) that the lead time L of the process will be 8 units of time.

Conversion of Bond Graphs to Signal Flow Graphs

Bond graphs for linear systems can be easily converted to signal flow graphs (SFG) and transfer functions between inputs and outputs can be obtained for further analysis using linear control theory. The procedure for such conversion is laid out here.

Nodes of SFG (Junction Equality Laws)

A signal flow graph constitutes of nodes that are connected by directed lines called branches. Each branch has an associated gain. Unlike bond graphs, SFGs do not have any causal and power information. The branches simply specify traversal path and are associated with only one signal. Each bond in a bond graph thus is represented by two nodes in a SFG (effort and flow node). Activated bonds are represented by only one node (e.g. a flow activated bond will be represented by an effort node and an effort activated bond by only a flow node).

All bonds connected to a 1-Junction have the same flow and are thus represented by a single flow node. Any flow activated bond at a 1-Junction is dropped from the SFG. Similarly, all bonds connected to a 0-junction are represented by a single effort node and effort activated nodes at that junction are not considered. The nomenclature for a flow node representing flow in bonds of a 1-junction is f_i,j,k.., where i,j,k.. are the bond numbers of bonds connected to that junction. Effort nodes for bonds at a 0-junction are similarly represented by e_i,j,k...

All bonds connected to a 1-junction contribute separate effort nodes (viz, e_i, e_j .., where i, j, .. are bond numbers) leaving apart those which are effort activated and those which have been already represented in SFG. Similarly, All non-flow activated bonds connected to a 0-junction contribute separate flow nodes (viz, f_i, f_j .., where i, j, .. are bond numbers) except those which have been already represented in SFG.

Branches and Gains (Elemental Relations)

For I-element in integral causality, the equation is f = m^-1ò e dt.
Taking Laplace transform of both sides, f(s) = e(s)/(m s) or f(s)/e(s) = 1/(m s).

Thus the gain associated with an integrally causalled I-element is 1/ms while the branch is directed from the effort to flow node (cause to effect). Similarly, for a differentially causalled I-element, the gain is ms and the branch is directed from the flow to effort node (cause to effect).

Cause and effect relation for an integrally causalled C-element is e = K ò f dt.
Taking Laplace transform of both sides, e(s) = K e(s)/s or e(s)/f(s) = K/s.

Thus the gain associated with an integrally causalled C-element is K/s while the branch is directed from the flow to effort node (cause to effect). For a differentially causalled C-element, the gain is s/K and the branch is directed from the effort to flow node (cause to effect).

The relationship between cause and effect for R-elements are not described by any differentiation or integration. Thus the gain term doesn't contain the Laplace variable s. The gain for R-element in resistive causality is e(s)/f(s) = R whereas; for conductive causality, it is f(s)/e(s) = 1/R.

Similar relationships can be established for two-ports TF and GY. The bond graph elements in different causal patterns and their corresponding signal flow graph representations are shown in the table below.

Bond Graph Signal Flow Graph

Receptors (Junction Algebra)

1-Junction is a flow equalizing and effort sum junction. The strong bond for the 1-Junction is the bond that has causality away from the junction. This strong bond thus provides information of flow to the junction. The weak variables of 1-Junction are efforts. The effort equation for the 1-junction is written for the weak variables (efforts), where the effort in the strong bond is expressed as signed sum of effort in other bonds. This weak effort variable of the strong bond is called the receptor of the junction. For the 1-junction shown below, the junction algebra equation is e2=e1-e3. The signal flow graph representation is shown to the right.

Bond number 2 is the strong bond and the weak variable e₂ is the receptor node. Signals from other nodes for weak variables are added to this node. The gain for all branches in SFG corresponding to bonds that are power directed in opposite direction as compared to the strong bond (i.e. contra-oriented) is 1. Otherwise, for all co-oriented bonds (bonds having same power orientation as the strong bond as seen from the junction) the gain is -1.

The receptor for a 0-junction is the weak flow variable of the strong bond ( the bond that decides effort of the 0-junction, i.e. causalled near the junction). For the 0-junction shown below, the junction algebra can be represented at the receptor node f2 as shown to the right.