06.Interconnections and feedback

This version: 22/10/2004

Chapter 6

Interconnections and feedback

We now begin to enter into the more “design” oriented parts of the course. The concept of feedback is central to much of control, and we will be employing the analysis tools developed in the time-domain, the s-domain, and the frequency domain to develop ways of evaluating and designing feedback control systems. The value of feedback is at the same time obvious and mysterious. It is clear that it ought to be employed, but it often has e ects that are subtle. Somehow the most basic feedback scheme is the PID controller that we discuss in Section 6.5. Here we can get an idea of how feedback can e ect a closed-loop system.

You may wish to take a look at the DC motor system we talked about in Section 1.2 in order to see a very concrete display of the disadvantages of open-loop control, and how this can be repaired to advantage with a closed-loop scheme.

Contents

6.5.1 Proportional control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.5.2 Derivative control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.5.3 Integral control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.5.4 Characteristics of the PID control law . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.1 Signal flow graphs

We have seen in our discussion of block diagrams in Section 3.1 that one might be interested in connecting a bunch of transfer functions like the one depicted in Figure 3.7. In this section we indicate how one does this in a systematic and general way. This will enable us to provide useful results on stability of such interconnected systems in Section 6.2.3. The advantage of doing this systematically is that we can provide results that will hold for any block diagram setup, not just a few standard ones.

The signal flow graph was first studied systematically in control by Mason[1953, 1956]. Many of the issues surrounding signal flow graphs are presented nicely in the paper of Lynch [1961]. A general discussion of applications of graph theory may be found in the book of Chen [1976]. Here the signal flow graph can be seen as a certain type of graph, and its properties are revealed in this context. Our presentation will often follow that of Zadeh and Desoer [1979].

6.1.1 Definitions and examples The notion of a signal flow graph can be seen as a modification of the concept of a block diagram The idea is to introduce nodes to represent the signals in a system, and then connect the nodes with branches, and assign to each branch a rational function that performs the duty of a block in a block diagram. For example, Figure 3.1 would simply appear as shown in Figure 6.1. Before we proceed to further

Figure 6.1 A simple signal flow graph

illustrations of how to construct signal flow graphs along the lines of what we did with block diagrams, let’s say just what we are talking about. Part of the point of signal flow graphs is that we can do this in a precise way.

6.1Definition Denote n = {1, 2, . . . , n} and let I n × n.

(i)A signal flow graph with interconnections I is a pair (S, G) where S is a collection {x1, . . . , xn} of nodes or signals, and

G = {Gij R(s) | (i, j) I}

is a collection of rational functions that we call branches or gains. The branch Gij originates from the node xj and terminates at the node xi.

(ii)A node xi is a sink if no branches originate from xi.

(iii)A node xi is a source if no branches terminate at xi.

(iv)A source xi is an input if only one branch originates from xi, and the gain of this branch is 1.

(v)A sink xi is an output if only one branch terminates at xi and this branch has weight

For example, for the simple signal flow graph of Figure 6.1, we have n = {1, 2}, I = {(1, 2)}, S = {x1, x2}, and G = {G21 = R}. The node x1 is a source and the node x2 is a sink. In this case note that we do not have the branch from x2 to x1. This is why we define I as we do—we will almost never want all n2 possible interconnections, and those that we do want are specified by I. Note that we assume that at most one branch can connect two given nodes. Thus, if we have a situation like that in Figure 6.2, we will replace this with the

Figure 6.2 Two branches connecting the same nodes

situation in Figure 6.3.

Figure 6.3 Branches added to give a single branch

If you think this obtuse and abstract, you are right. But if you work at it, you will see why we make the definitions as we do. Perhaps a few more examples will make things clearer.

6.2Examples 1. The signal flow graph corresponding to the series block diagram of Figure 3.2 is shown in Figure 6.4. Note that here we have n = {1, 2, 3}, I = {(1, 2), (2, 3)},

Figure 6.4 The signal flow graph for a series interconnection

S = {x1, x2, x3}, and G = {G21, G32}. The node x1 is a source and x3 is a sink. There are a possible n2 = 9 interconnections, and we only have two of them realised in this graph. The transfer function from x1 to x2 is simply G21; that is, x2 = G21x1. We also read o

x3 = G32x2, and so x3 = G21G32x3.

2.We now look at the representation of a parallel interconnection. The signal flow graph is shown in Figure 6.5, and we have n = {1, 2, 3, 4}, I = {(1, 2), (1, 3), (2, 4), (3, 4)}, S = {x1, x2, x3, x4}, and G = {G21, G31, G42 = 1, G43 = 1}. One sees that x1 is a source and x4 is a sink. From the graph we read the relations x2 = G21x1, x3 = G31x1, and x4 = x2 + x3. This gives x4 = (G21 + G31)x1.

3.The preceding two signal flow graphs are very simple in some sense. To obtain the transfer function from the signal flow graph is a matter of looking at the graph and applying the obvious rules. Let us now look at a “feedback loop” as a signal flow graph. The graph

Figure 6.6 The signal flow graph for a negative feedback loop

is in Figure 6.6, and we have n = {1, 2, 3, 4, 5}, I = {(1, 2), (2, 3), (3, 4), (4, 2), (4, 5)}, S = {x1, x2, x3, x4, x5}, and G = {G21 = 1, G32, G43, G24 = −1, G54 = 1}. Clearly, x1 is an input and x5 is an output. From the graph we read the relationships x2 = x1 − x4, x3 = G32x2, x4 = G43x3, and x5 = x4. This gives, in the usual manner,

4. Suppose we wish to extract more information from the negative feedback loop of the previous example. In Figure 6.7 we depict a situation where we have added an input

x6

Figure 6.7 Signal flow graph for negative feedback loop with extra structure

signal x6 to the graph, and tapped an output signal x7 = x2. Thinking about the control situation, one might wish to think of x6 as a disturbance to the system, and of x7 as being the error signal (this will be seen in a better context in Sections 6.3, 8.3, and 8.4). In doing so we have added the input x6 to the existing input x1 and the output x7 to the existing output x5. Let us see how the two outputs get expressed as functions of the two inputs. We have the relations x2 = x1 − x4, x3 = G32x2 + x6, x4 = G43x3, x5 = x4, and

We see that we essentially obtain a system of two linear equations that expresses the input/output relations for the graph.

One can consider any node xj to e ectively be an output by adding to the signal flow graph a node xn+1 and a branch Gn+1,j with gain equal to 1. One can also add an input to any node xj by adding a node xn+1 and a branch Gj,n+1 whose gain is 1.

6.1.2 Signal flow graphs and systems of equations A signal flow graph is also a representation of a set of linear equations whose coe cients belong to any field. That is to say, we consider a set of linear equations where the coe cients are anything that can be added, multiplied, and divided. The particular field that is of interest to us is R(s). What’s more, we consider a very particular type of linear equation; one of the form

−Gn1x1 − Gn2x2 − · · · + (1 − Gnn)xn = un,

where Gij R(s), i, j = 1, . . . , n. The reason for using this type of equation will become clear shortly. However, we note that corresponding to the equations (6.1) is a natural signal flow graph. The following construction indicates how this is determined.

6.3 From linear equation to signal flow graph Given a set of linear equations of the form (6.1), perform the following steps:

(i)place a node for each variable x1, . . . , xn;

(ii)place a node for each input u1, . . . , un;

(iii)for each nonzero Gij, i, j = 1, . . . , n, draw a branch originating from node j and terminating in node i, having gain Gij;

(iv)for i = 1, . . . , n, draw a branch of gain 1 from ui to xi.

The result is a signal flow graph with nodes {x1, . . . , xn, xn+1 = u1, . . . , x2n = un} and gains

G = {Gij | i, j = 1, . . . , n} {G1,n+1 = 1, . . . , Gn,2n = 1}.

For example, in Figure 6.8 we show how this is done when n = 2. One can readily verify that a “balance” at each node will yield the equations (6.1).

This establishes a graph for each set of equations of the form (6.1). It is also true that one can go from a graph to a set of equations. To state how to do this, we provide the following recipe.

Figure 6.8 A general 2 node signal flow graph

6.4 From signal flow graph to linear equation Given a signal flow graph (S, G), to any source xi that is not an input, add a node xji = ui to S and a branch Giji to G. After doing this for each such source, we arrive at a new signal flow graph (S0, G0). For each node xi in (S0, G0) that is not an input perform the following steps:

(i)let xj1 , . . . , xjk be the collection of nodes for which there are branches connecting them with xi;

(ii)form the product of xj1 , . . . , xjk with the respective branch gains;

(iii)set the sum of these products equal to xi.

The result is an equation defining the node balance at node xi. Some of the inputs may be zero.

Thus we establish a 1−1 correspondence between signal flow graphs and linear equations of the form (6.1) (with some inputs and gains possibly zero). Let us denote by GS,G the matrix of rational functions that serves as the coe cient matrix in (6.1). Thus

We call GS,G the structure matrix for (S, G). Note that GS,G is a matrix whose components are rational functions. We denote the collection of n × n matrices with rational function components by R(s)n×n. Again we note that for a given signal flow graph, of course, many of the terms in this matrix might be zero. The objective of making the connection between signal flow graphs and linear equations is that the matrix formulation puts at our disposal all of the tools from linear algebra. Indeed, one could simply use the form (6.1) to study signal flow graphs. However, this would sweep under the carpet the special structure of the equations that results from their being derived as node equations of a signal flow graph. One of the main objectives of this section is to deal with this aspect of the signal flow graph. But for now, let us look at our signal flow graphs of Example 6.2 as systems of equations.

6.5 Examples (Example 6.2 cont’d) We shall simply write down the matrix G that appears in (6.1). In each case, we shall consider, as prescribed by the above procedure, the system with an input attached to each source that is not itself an input. Thus we ensure that the matrix GS,G represents all states of the system.

to have a good understanding of paths through a signal flow graph. To do this, we make some definitions.

6.6Definition Let (S, G) be a signal flow graph with interconnections I n × n.

(i)A subgraph of (S, G) is a pair (S0, G0) where S0 S and where G0 G satisfies

Gij G0 implies that xi, xj S0. If S0 = {xi1 , . . . , xin0 }, then we say (S0, G0) has interconnections {i1, . . . , in0 }. We also denote by I0 I the subset defined by G0. Thus (i0, j0) I0 if and only if Gi0j0 G0.

(ii)A subgraph (S0, G0) is connected if for any pair xi, xj S0 there exists nodes xi0 =

xi, xi1 , . . . , xik = xj S0 and gains G1, . . . , Gk G0 so that G` {Gi`,i`+1 , Gi`+1,i` },

` {0, . . . , k − 1}.

(iii)A path in (S, G) is a sequence {xi1 , . . . , xik } of nodes with the property that (ij, ij+1) I for each j {1, . . . , k−1}. That is to say, there is a branch connecting the jth element in the sequence with the (j + 1)st element in the sequence.

(iv)A path {xi1 , . . . , xik } is simple if the set {(ij+1, ij) | j {1, . . . , k − 1}} is distinct.

(v)A path {xi1 , . . . , xik } is directed if the vertices are distinct. We denote the set of all directed paths in (S, G) by Path(S, G).

(vi)A forward path is a directed path from an input node of (S, G) to a node of (S, G). The set of forward paths from an input xi to a node xj is denoted Pathji(S, G).

(vii)The number of branches in a directed path is the length of the directed path.

(viii)A loop is a simple path {xi1 , . . . , xik } with the property that xi1 = xik . The set of loops in (S, G) we denote by Loop(S, G).

(ix)The product of the gains in a directed path is the gain of the directed path.

(x)The loop gain of a loop L is the product of the gains comprising the loop, and is denoted GL.

(xi)A finite collection of loops is nontouching if they have no nodes or branch gains in

Perhaps a few words of clarification about the less obvious parts of the definition are in order.

1.The notion of a connected subgraph has a simplicity that belies its formal definition. It merely means that it is possible to go from any node to any other node, provided one ignores the orientation of the branches.

2.Note that the nodes in a loop must be distinct. That a loop cannot follow a path that goes through any node more than once.

3.It is easy to be misled into thinking that any directed path is a forward path. This is not necessarily true since for a forward path originates at an input for (S, G).

Again, this looks pretty formidable, but is in fact quite simple. We can illustrate this easily by looking at the examples we have given of signal flow graphs.

6.7Examples 1. For the very simple signal flow graph of Figure 6.1 there is but one path of length 1, and it is {x1, x2}. There are no loops.

2. For the series signal flow graph of Figure 6.4 we have two paths of length 1,

{x1, x2}, {x2, x3},

and one path of length 2,

{x1, x2, x3}.

Again, there are no loops.

3.The parallel signal flow graph of Figure 6.5, forgetting for the moment that some of the gains are predetermined to be 1, has the paths

{x1, x2}, {x2, x4}, {x1, x3}, {x3, x4},

of length 1, and paths

{x1, x2, x4}, {x1, x3, x4}

of length 2. There are no loops in this graph either.

4.The negative feedback graph of Figure 6.7 has an infinite number of paths. Let us list the basic ones.

(a)length 1:

{x1, x2}, {x2, x3}, {x3, x4}, {x4, x5}, {x3, x6}, {x2, x7}.

(b) length 2:

{x1, x2, x3}, {x2, x3, x4}, {x3, x4, x5}, {x3, x4, x6}, {x2, x4, x7}.

(c) length 3:

{x1, x2, x3, x4}, {x2, x3, x5, x5}, {x3, x4, x5, x6}.

(d) length 4:

{x1, x2, x3, x4, x5}.

Some of these paths may be concatenated to get paths of any length desired. The reason for this is that there is a loop given by

{x2, x3, x4}.

6.1.4 Cofactors and the determinants Next we wish to define, and state some properties of, some quantities associated with the matrix GS,G. These quantities we will put to use in the next section in proving an important theorem in the subject of signal flow graphs: Mason’s Rule. This is a rule for determining the transfer function between any input and any output of a signal flow graph. But this is getting ahead of ourselves. We have a lot of work to do in the interim.

Let us proceed with the development. For k ≥ 1 we denote

Loopk(S, G) = {(Lj1 , . . . , Ljk ) (Loop(S, G))k | L1, . . . , Lk are nontouching}.

That is, Loopk(S, G) consists of those k-tuples of loops, none of which touch the others. Note that Loop1(S, G) = Loop(S, G) and that for k su ciently large (and not very large at all in

220 6 Interconnections and feedback 22/10/2004

most examples we shall see), Loopk(S, G) = . The determinant of a signal flow graph (S, G) is defined to be

It turns out that S,G is exactly the determinant of GS,G.

6.8 Proposition S,G = det GS,G.

Proof The proof is involved, but not di cult. We accomplish it with a series of lemmas.

α

where Gα is a product of branch gains. We denote by (Sα, Gα) the subgraph of (S, G) comprised of the nodes and branches that are involved in a typical term Gα.

We explore some properties of the subgraph (SG, GG).

1 Lemma For each node xi Sα, there is at most one branch in Gα terminating at xi, and at most one branch in Gα originating from xi.

Proof Let G = {Gα1, . . . , Gαk} be the gains that form Gα. By the definition of the determinant, for each row of GS,G there is at most one j {1, . . . , k} so that Gαj is an element of that row. A similar statement holds for each column of GS,G. However, from these statements exactly follows the lemma. H

2 Lemma Gα is a product of loop gains of nontouching loops.

Proof Suppose that (Sα, Gα) consists of a collection (Sα1, Gα1), . . . , (Sα`, Gα`) of connected subgraphs. We shall show that each of these connected subgraphs is a loop. It then follows from Lemma 1 that the loops will be nontouching. We proceed by contradiction, supposing that one of the connected components, say (Sα1, Gα1), is not a loop. From Lemma 1 it follows that (Sα1, Gα1) is a directed path between two nodes; suppose that these nodes are xi and xj. Since no branches terminate at xi, there are no elements from row i of GS,G in Gα. Since no branches originate from xj, there are no elements from column j of GS,G in Gα. It therefore follows that the 1’s in position (i, i) and (j, j) appear in the expression for Gα. However, since Sα1, Gα1) is a directed path between xi and xj it follows that there are some ki, kj n

Proof First note that the determinant of GS,G is una ected by the numbering of the nodes. Indeed, if one interchanges i and j in a given numbering scheme, then in GS,G, both the ith and jth columns and the ith and jth rows are swapped. Since each of these operations gives a change of sign in the determinant, the determinant itself is the same.

Suppose that (Sα, Gα) is comprised of k nontouching loops of lengths `1, . . . , `k. Denote the nodes in these loops by

{x1, . . . , x`1 , x1}, {x`1+1, . . . , x`2 , x`+1}, . . . , {x`k−1+1, . . . , x`k , x`k−1+1}.

According to the definition (A.1) of the determinant, to determine the sign of the contribution of Gα, we should determine the sign of the permutation

By `1 −1 transpositions we may shift the 1 in the `1st position to the 1st position. By `2 −1 transpositions, the `1 + 1 in the (`1 + `2)th position can be shifted to the (`1 + 1)st position. Proceeding in this manner, we see that the sign of the permutation is given by

The three previous lemmas show that the terms Gα in the expression (6.4) for det GS,G have the form of the terms in S,G. It remains to show that every term in G,G appears in det GS,G. Thus suppose that (Sα1, Gα1), . . . , (Sα`, Gα`) is a collection of nontouching loops. Since these loops are nontouching, it follows that in any given row or column, there can reside at most one gain from the set Gα1 . . . Gα`. Therefore, an expansion of the determinant will contain the product of the gains from the set Gα1 . . . Gα`. Furthermore, they must have the same sign in det GS,G as in S,G by Lemma 3. This concludes the proof.

Let us see how to apply this in two examples, one of which we have seen, and one of which is complicated enough to make use of the new terminology we have introduced.

6.9Examples 1. Let us first look at the signal flow graph depicted in Figure 6.7. For clarity it helps to ignore the fact that G24 = −1. This graph has a single loop

L = {x2, x3, x4, x2}.

This means that Loop1(S, G) = {L} and Loopk(S, G) = for k ≥ 2. The gain of the loop is simply the product G32G43G24. Therefore, in this case, the determinant as per (6.3) is

S,G = 1 − G32G43G24.

If we now remember that G24 = −1 we get the term 1 + G32G43 which appears in the denominators in Example 6.2–4. This, of course, is no coincidence.

Figure 6.12 A signal flow graph with multiple loops

2.Next we consider a new signal flow graph, namely the one depicted in Figure 6.12. This graph has four loops:

L1 = {x2, x3, x2}, L2 = {x3, x4, x3}, L3 = {x6, x7, x6}, L4 = {x7, x8, x7}.

We have Loop1(S, G) = {L1, L2, L3, L4} and

Loop2(S, G) = {(L1, L3), (L1, L4), (L2, L3), (L2, L4), (L3, L1), (L4, L1), (L3, L1), (L4, L2)}.

An application of (6.3) gives

S,G = 1 − (GL1 + GL2 + GL3 + GL4 ) + (GL1 GL3 + GL1 GL4 + GL2 GL3 + GL2 GL4 ).

One may, of course, substitute into this expression the various gains given in terms of the branch gains.

Now we turn to the cofactor of a path. For P Path(S, G) let GP denote the branches of G with those comprising P removed, and those branches having a node in common with P removed. We note that (S, GP ) is itself a signal flow graph. If (S, G) is connected, then (S, GP ) may not be. For P Path(S, G) the cofactor of P is defined by CofP (S, G) = S,GP .

Let us illustrate this for the examples whose determinants we have computed.

6.10 Examples 1. For the signal flow graph depicted in Figure 6.7 let us consider various paths. Again we do not pay attention to specific values assign to branch gains.

(a)P1 = {x1, x2, x3, x4, x5}: If we remove this path, the graph has no loops and so CofP1 (S, G) = 1.

(b)P2 = {x1, x2, x7}: Removing this leaves intact the existing loop, and so the deter-

minant of the graph remains unchanged. Therefore we have CofP2 (S, G) = S,G =

1 − G32G43G24.

(c)P3 = {x3, x4, x5, x6}: Removal of this path leaves a signal flow graph with no loops so we must have CofP3 (S, G) = 1.

(d)P4 = {x2, x3, x4, x6, x7}: Again, if P4 is removed, we are left with no loops and this then gives CofP4 (S, G) = 1.

2.Next we look at the signal flow graph of Figure 6.12. We consider two paths.

(a)P1 = {x1, x2, x3, x4, x5}: Removing this loop leaves two loops remaining: L3 and L4. The determinant of the resulting graph is, by (6.3),

CofP1 (S, G) = 1 − (GL3 + GL4 ).

(b)P2 = {x1, x6, x7, x8, x5}: This situation is rather like that for the path P1 and we determine that

6.1.5 Mason’s Rule With the notion of cofactor and determinant clearly explicated, we can state Mason’s Rule for finding the transfer function between an input to a signal flow graph and any node in the graph. In this section we suppose that we are working with a signal flow graph (S, G) with interconnections I n × n. In order to simplify matters and make extra hypotheses for everything we do, let us make a blanket assumption in this section.

The following result gives an easy expression for the transfer function between the input ui at xi and any other node. The following result can be found in the papers of Mason [1953, 1956]. It is not actually given a rigorous proof there, but one is given by Zadeh and Desoer [1979]. We follow the latter proof.

6.11 Theorem (Mason’s Rule) Let (S, G) be a signal flow graph. For i, j {1, . . . , n} we have

Proof Without loss of generality, suppose that i = 1 and that node 1 is an input. Denote x1 = (x1, 0, . . . , 0) and let GS,G(x1, j) be the matrix GS,G with the jth column replaced with x1. Cramer’s Rule then says that

xj = det GS,G(x1, j).

From Proposition 6.8 the theorem will follow if we can show that

Path (S,G)

Let (Sj, Gj) be the subgraph of (S, G) corresponding to the matrix GS,G(x1, j). One can readily ascertain that one may arrive at (Sj, Gj) from (S, G) by performing the following operations:

1.remove all branches originating from xj;

2.add a branch with gain x1 originating from xj and terminating at x1.

Since the only nonzero term in the jth column is the term x1 appearing in the first row, an expansion of the determinant about the jth column shows that det GS,G(x1, j) has the form

Y det GS,G(x1, j) = x1 Gα.

α

By Proposition 6.8 we know that det GS,G(x1, j) is comprised of a sum of products of loop gains for nontouching loops. Thus in each of the products must appear a loop gain of the form

f1Gk21Gk3k2 · · · Gjk`−1 .

It follows that det GS,G(x1, j) is a sum of products of terms falling into three types:

1.x1;

2.the gain of a forward path P from x1 to xj;

3.a product of loop gains for loops that share no nodes or branches with P .

Let us fix such a term corresponding to a given forward path Pα. Now let us renumber the nodes so that we have

Now note the form of the matrix in (6.6) shows that the signal flow graph corresponding

˜

to G is obtained from (S, G) by removing all branches associated with the forward path Pα. From this it follows that all products in det GS,G(x1, j) are of the form GP CofP (S, G) for some forward path P .

It remains to show that any such expression will appear as a product in the expression for det GS,G(x1, j). This may be shown as follows. Let P be a forward path comprised of gains Gk21, Gk3k2 , . . . , Gj,k`−1 . The structure of (Sj, Gj) implies that by adding the gain x1, we have the gain for a loop in (Sj, Gj). Now, as we saw in the proof of Proposition 6.8, this term, multiplied by the loop gains of nontouching loops, is ensured to appear in the determinant of (Sj, Gj).

If a signal flow graph has multiple inputs u1, . . . , uk, then one can apply Mason’s rule for each input, and the resulting expression for a non-input node xj is then of the form

is the graph transmittance from the input ui to the node xj. Note that it is possible that for a given i and j, Pathji(S, G) will be empty. In this case, we take the graph transmittance to be zero.

The following result gives a useful interpretation of the set of all graph transmittances.

22/10/2004 6.1 Signal flow graphs 225

Proof As we saw in the proof of Theorem 6.11, the numerator term in the expression for Tji is the determinant of the matrix obtained by replacing the jth column of GS,G by the ith standard basis vector ei. By Cramer’s Rule we infer that Tji is the jth component of the solution vector ti for the linear equation GS,Gti = ei. This means that if we define

T S,G = t1 · · · tn ,

then we have

Let us show how the above developments all come together to allow an easy determination of the various transfer functions for the two examples we are considering.

6.13 Examples (Example 6.9 cont’d) In each example, we will number paths as we did in Example 6.9.

1.For the signal flow graph of Figure 6.7 there are two sinks, x1 and x6, and two sources, x5 and x7. To each of the sources, let us attach an input so that we are properly in the setup of Theorem 6.11. Recalling our labelling of paths from Example 6.9, we have

These are, of course, the paths whose cofactors we computed in Example 6.9. Now we can compute, using Mason’s Rule, the coe cients in expressions of our two sinks x5 and x7 involving the two sources x1 and x6. We have

T51 = G21G32G43G54

1 − G32G43G24

G21G72

T71 = 1 − G32G43G24

G36G43G54

T56 = 1 − G32G43G24

T76 = G36G43G24G72

1 − G32G43G24

and so we thus obtain the explicit expressions

x5 = T51x1 + T56x6, x7 = T71x1 + T76x6.

Let’s see how this checks out when G21 = G54 = G72 = G36 = −G24 = 1. In this case we obtain

as we did when we performed the calculations “by hand” back in Example 6.2.

2.We shall compute the transfer function from x1 to x5. We have Path51(S, G) = {P1, P2}. By Mason’s Rule, and using the determinant and cofactors we have already computed,

we have

We have provided in this section a systematic way of deriving the transfer function between various inputs and outputs in a signal flow graph. What’s more, we have identified an important piece of structure in any such transfer function: the determinant of the graph. We shall put this to use when studying stability of interconnected systems in Section 6.2.3.

6.1.6 Sensitivity, return di erence, and loop transmittance Up to this point, the discussion has been centred around the various transfer functions appearing in a signal flow graph. Let us now look at other interesting objects, sensitivity, return di erence, and loop transmittance. These will turn out to be interesting in the special context of single-loop systems in Section 6.3. The ideas we discuss in this section are presented also in the books of Truxal [1955] and Horowitz [1963].

First we consider the notion of sensitivity. Let us first give a precise definition.

6.14 Definition Let (S, G) be a signal flow graph with interconnections I n×n, let i, j n, and let (`, k) I. Let Tji be the graph transmittance from node i to node j and let G`k be the branch gain from node k to node `. The sensitivity of Tij to Gk` is

Let us try to gather some intuition concerning this definition. If f is a scalar function of a scalar variable x then note that

1d(f(eln x))

by f(x0), with respect to x, normalised by x0. Thus one might say that

d(ln f(x)) = d(% change in f). d(ln x) d(% change in x)

In any event, S`kji measures the dependence of Tji on G`k in some sense. Let us now give a formula for S`kji in terms of graph determinants.

22/10/2004 6.1 Signal flow graphs 227

6.15 Proposition Let (S, G) be a signal flow graph with interconnections I n×n, let i, j n, and let (`, k) I. Let G(`,k) = G \ {G`k}. We then have

It is much easier to say in words what the symbols in the result mean. The signal flow graph (S, G(`,k)) is that obtained by removing the branch from node k to node `. Thus the numerators,

in each of the terms on the right-hand side of (6.7) are simply the numerator and denominator for the transfer function from node i to node j in the graph (S, G(`,k)) as given by Mason’s rule. Thus these can often be obtained pretty easily. Let us see how this works in an example.