2.2 The block diagram. Figure 2-3-1 shows the block diagram of a linear feedback control system The following terminology often used in control systems is defined with preference to the block diagram R(S),r(t=reference input C(s),c(t=output signal(controlled variable) B(s, b(t=feedback signal E(S),e(t=R(s-C(s=error signal G(s=C(s)/c(s=open-loop transfer function or forward-path transfer function MS=C(S/R(S=closed-loop transfer function H(S=feedback-path transfer function G(SH(S=loop transfer function G(s) S Fig2-2-1 2022-2-3 16
2022-2-3 16 2.2 The block diagram. Figure 2-3-1 shows the block diagram of a linear feedback control system. The following terminology often used in control systems is defined with preference to the block diagram. R(s), r (t)=reference input. C(s), c (t)=output signal (controlled variable). B(s), b (t)=feedback signal. E(s), e (t)=R(s)-C(s)=error signal. G(s)=C(s)/c(s)=open-loop transfer function or forward-path transfer function. M(s)=C(s)/R(s)=closed-loop transfer function H(s)=feedback-path transfer function. G(s)H(s)=loop transfer function. G(s) H(s) Fig2-2-1
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) 2022-2-3 17
2022-2-3 17 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 2022-2-3 18
2022-2-3 18 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 42 21 12 A 2022-2-3 19
2022-2-3 19 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
2022-2-3 20 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