The closed -loop transfer function can be expressed as a function of G(s)and H(s). From Fig. 2-2-1we write C(S=G(Sc(s) (2-2) B(S=H(SC(S) 2-3) The actuating signal is written C(S=R(S-B(S) Substituting eq(2-4 )into eq(2-2)yields C(S=G(SR(S-G(SB(S) Substituting eq(2-3)into eq(2-5)gives C(S=G(SR(S)G(SH(SC(S) 2-6) Solving C(s) from the last equation the closed-loop transfer function of the system is given by M(s)=C(S)R(s)=G(s)/(1+G(s)H(s) (2-7)
The closed –loop transfer function can be expressed as a function of G(s) and H(s). From Fig.2-2-1we write: C(s)=G(s)c(s) (2-2) B(s)=H(s)C(s) (2-3) The actuating signal is written C(s)=R(s)-B(s) (2-4) Substituting Eq(2-4)into Eq(2-2)yields C(s)=G(s)R(s)-G(s)B(s) (2-5) Substituting Eq(2-3)into Eq(2-5)gives C(s)=G(s)R(s)-G(s)H(s)C(s) (2-6) Solving C(s) from the last equation ,the closed-loop transfer function of the system is given by M(s)=C(s)/R(s)=G(s)/(1+G(s)H(s)) (2-7)
2.3 Signal flow graphs Fundamental of signal flow graphs A simple signal flow graph can be used to represent an algebraic relation It is the relationship between node i to node with the transmission function A, (it is also represented by a branch) A.·X (2-8 Node A Node X Branch
2.3 Signal flow graphs ¨ Fundamental of signal flow graphs A simple signal flow graph can be used to represent an algebraic relation It is the relationship between node i to node with the transmission function A, (it is also represented by a branch). X i A ij X j (2-8)
2.3.1 Definitions Let us see the signal flow graphs
2.3.1 Definitions ¨ Let us see the signal flow graphs
Definition 1: A path is a Continuous, Unidirectional Succession of branches along which no node is passed more than once. For example, x, to x, to X, to X4 X2,Y, and back to x, and x, to x, to x4 are paths Definition 2 An Input Node Or Source is a node with only outgoing branches. For example, x is an input node Definition 3: An Output Node or sink is a node with only A
Definition 1: A path is a Continuous, Unidirectional Succession of branches along which no node is passed more than once. For example, to to to , and back to and to to are paths. X1 X 2 X 3 X 4 2 3 X , X X 2 X1 X 2 X 4 Definition 2: An Input Node Or Source is a node with only outgoing branches. For example, X1 is an input node. Definition 3: An Output Node Or Sink is a node with only incoming branches. For example, is an output node. X 4