Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Exercises(6) 2 —14, 15, 17, 18 Automatic Control Theory

2.3 Mathematical Model in Complex Domain – Transfer Function （ 1 ） Definition, properties and its application scope （ 2 ） Transfer Functions for Common Components （ 3 ） Typical Factors 2.4 Block Diagrams of Control Systems （ 1 ） Constructing the block diagram of control systems （ 2 ） Block diagram reduction 2.3 Mathematical Model in Complex Domain – Transfer Function （ 1 ） Definition, properties and its application scope （ 2 ） Transfer Functions for Common Components （ 3 ） Typical Factors 2.4 Block Diagrams of Control Systems （ 1 ） Constructing the block diagram of control systems （ 2 ） Block diagram reduction Review

Automatic Control Theory （ Lecture 6 ） Chapter 2 Mathematical Models of Control Systems §2.1 Introduction §2.2 Time-Domain Mathematical Models of Control Systems §2.3 Complex Domain Mathematical Models of Control Systems §2.4 Block Diagrams of Control Systems §2.5 Signal Flow Graphs of Control Systems §2.6 Transfer Functions of control systems

Mathematical model of control system

§2.5 Signal Flow Graphs of Control Systems §2.5.1 The corresponding relation between the signal flow graph and the block diagram Signal flow graphStructure diagram Source node 源节点 Input signal Sink node 阱节点 Output signal Mixed node 混合节点 Summing Point Pickoff Point Branch 支路 Block Branch gain 支路增益 Transfer function of a block Forward path 前向通路 Loop 回路 Nontouching loop 互不接触回路

The conversion between signal flow graph and block diagram （ 1 ） Signal flow graph of control system

（ 1 ） Signal  Structure diagram Structure diagram of control system

The conversion between signal flow graph and structure diagram （ 1 ） Structure diagram of control system

（ 2 ） Structure diagram  Signal flow graph Signal flow graph of system

Mason ’ s gain formula: — determinant of graph 特征式 — umber of forward path 前向通路的条数 — path gain of kth forward path 第 k 条前向通路的总增益 — sum of all individual loop gains 所有不同回路的回路增益之和 — sum of gain products of all possible combinations of two nontouching loops 两两互不接触回路的回路增益乘积之和 — sum of gain products of all possible combinations of three nontouching loops 互不接触回路中，每次取其中三个的回路增益 乘积之和 — cofactor of the kth forward path determinant of the graph with the loops touching the kth forward path removed 第 k 条前向通路的余子 式 ( 把与第 k 条前向通路接触的回路去除，剩余回路构成的子特征式 §2.5.2 Mason’s gain formula

Mason’s Fomula （ 1 ） Example 1 Obtain the transfer function C(s)/R(s) The structure diagram of control system

Example 1 Determine C(s)/R(s)

Mason Formula （ 2 ） Example 2 Obtain the transfer function C(s)/R(s) The structure diagram of control system

Example 2 Determine C(s)/R(s)

Mason ’ s Fomula （ 3 ） Example 3 Determine transfer function C(s)/R(s) The structure diagram of control system

Example 3 Determine C(s)/R(s)

Mason ’ s fomula （ 4 ） Example 4 Obtain transfer function C(s)/R(s) The structure diagram of control system

Example 4 Determine C(s)/R(s)

Mason fomula （ 5 ） Example 5 Obtain transfer function C(s)/R(s) The block diagram of control system

Example 5 Determine C(s)/R(s)

Mason ’ s Fomula （ 6 ） Example 6 Obtain transfer function C(s)/R(s), C(s)/N(s) The block diagram of control system

Example 6 Determine C(s)/R(s), C(s)/N(s)

§2.6 Transfer function of control system 1.Transfer function of open loop 2.Transfer function of closed loop with the effect of the input r(t)

Transfer function of control system 3. Transfer function of closed loop with the effect of the disturbance n(t) 4. The output C(s) and error E(s) of system

Transfer function of control system (Example) Example 7 The structure diagram of system as shown in the right figure input r(t) = 1(t) disturbance n(t) =  (t) initial condition c(0) = -1 c ’ (0) = 0 determine the output c(t) and error e(r) Solution

Chapter 2 Summary

Excises (7) 2 —14, 15, 17, 18 Automatic Control Theory