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