《自动化仪表与过程控制》课程学习资料：Mathematical Foundation

2. Mathematical foundation

2.Mathematical Foundation

2.1 The transfer function concept From the mathematical standpoint algebraic and the dynamic behavior of a system In systems theory, the o differential or difference equations can be used to describe block diagram is often used to portray system of all types For linear systems, transfer functions and signal flow graphs are valuable tools for analysis as well as for design o If the input-output relationship of the linear system of Fig 1-2-1 is known, the characteristics of the system itself are also known The transfer function of a system is the ratio of the transformed output to the transformed input

2.1 The transfer function concept ¨ From the mathematical standpoint, algebraic and differential or difference equations can be used to describe the dynamic behavior of a system .In systems theory, the block diagram is often used to portray system of all types .For linear systems, transfer functions and signal flow graphs are valuable tools for analysis as well as for design ¨ If the input-output relationship of the linear system of Fig.1-2-1 is known, the characteristics of the system itself are also known. ¨ The transfer function of a system is the ratio of the transformed output to the transformed input

p output system p output TF(S) Finger 1-2-1 input-output relationships(a) general(b)transfer function TF(S) outputs) d inputs

system input output a TF(s) input output b ( ) ( ) ( ) ( ) ( ) r s c s inputs outputs TF s   Finger 1-2-1 input-output relationships (a) general (b) transfer function (2-1)

Summarizing over the properties of a function we state 1. a transfer function is defined only for a linear system, and strictly only for time-invariant system 2. A transfer function between an input variable and output variable of a system is defined as the ratio of the Lap lace transform of the output to the input 3. All initial conditions of the system are assumed to zero 4. a transfer function is independent of input excitation

Summarizing over the properties of a function we state: 1.A transfer function is defined only for a linear system, and strictly, only for time-invariant system. 2.A transfer function between an input variable and output variable of a system is defined as the ratio of the Lap lace transform of the output to the input. 3.All initial conditions of the system are assumed to zero. 4.A transfer function is independent of input excitation

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

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)

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

Definition 4 A Forward Path is a path from the input node to the output node. For example, x, to X2 to X to X4and X, to X2 to x are forward paths Definition 5: A Feedback Path or feedback loop is a path which originates and terminates on the same node. For example, x2 to x, and back to X2 is a feedback path Definition 6: A Self-Loop is a feedback loop consisting of a single branch. For example, A33 is a self-loop

Definition 4: A Forward Path is a path from the input node to the output node. For example, to to to and to to are forward paths. Definition 5: A Feedback Path or feedback loop is a path which originates and terminates on the same node. For example, to , and back to is a feedback path. Definition 6: A Self-Loop is a feedback loop consisting of a single branch. For example, is a self-loop. X1 X 2 X 3 X 4 X1 X 2 X 4 X 2 X 3 X 2 A33