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Robust Control Design with MATLAB
1 Introduction ............................................... 3
1.1 Control-system Representations . .......................... 4
1.2 System Stabilities . . . .................................... 6
1.3 Coprime Factorisation and Stabilising Controllers . ........... 7
1.4 Signals and System Norms . . .............................. 9
1.4.1 Vector Norms and Signal Norms . .................... 9
1.4.2 System Norms .................................... 10
2 Modelling of Uncertain Systems ........................... 13
2.1 Unstructured Uncertainties ............................... 13
2.2 Parametric Uncertainty .................................. 17
2.3 Linear Fractional Transformations . . . ...................... 20
2.4 Structured Uncertainties . . ............................... 23
3 Robust Design Specifications .............................. 25
3.1 Small-gain Theorem and Robust Stabilisation ............... 25
3.2 Performance Consideration ............................... 28
3.3 Structured Singular Values . . . ............................ 29
4 H∞ Design ................................................ 35
4.1 Mixed Sensitivity H∞ Optimisation. . ...................... 35
4.2 2-Degree-Of-Freedom H∞ Design . . ........................ 38
4.3 H∞ Suboptimal Solutions . ............................... 39
4.3.1 Solution Formulae for Normalised Systems ............ 39
4.3.2 Solution to S-over-KS Design....................... 43
4.3.3 The Case of D22 =0............................... 44
4.3.4 Normalisation Transformations . . .................... 45
4.3.5 Direct Formulae for H∞ Suboptimal Central Controller 47
4.4 Formulae for Discrete-time Cases . . ........................ 50xii Contents
5 H∞ Loop-shaping Design Procedures ...................... 55
5.1 Robust Stabilisation Against Normalised Coprime Factor
Perturbations . . ......................................... 56
5.2 Loop-shaping Design Procedures .......................... 58
5.3 Formulae for the Discrete-time Case ....................... 61
5.3.1 Normalised Coprime Factorisation of Discrete-time
Plant . . .......................................... 61
5.3.2 Robust Controller Formulae . ....................... 62
5.3.3 The Strictly Proper Case ........................... 63
5.3.4 On the Three DARE Solutions ...................... 65
5.4 A Mixed Optimisation Design Method with LSDP ........... 67
6 µ-Analysis and Synthesis .................................. 71
6.1 Consideration of Robust Performance ...................... 71
6.2 µ-Synthesis: D-K Iteration Method . ....................... 74
6.3 µ-Synthesis: µ-K Iteration Method . ....................... 77
7 Lower-order Controllers ................................... 79
7.1 Absolute-error Approximation Methods . ................... 80
7.1.1 Balanced Truncation Method ....................... 81
7.1.2 Singular Perturbation Approximation . . .............. 82
7.1.3 Hankel-norm Approximation . . ...................... 83
7.1.4 Remarks ......................................... 85
7.2 Reduction via Fractional Factors . ......................... 86
7.3 Relative-error Approximation Methods . . . .................. 90
7.4 Frequency-weighted Approximation Methods . . .............. 92
Part II Design Examples
8 Robust Control of a Mass-Damper-Spring System .........101
8.1 System Model ...........................................101
8.2 Frequency Analysis of Uncertain System....................107
8.3 Design Requirements of Closed-loop System ................108
8.4 System Interconnections . . ................................112
8.5 Suboptimal H∞ Controller Design . ........................115
8.6 Analysis of Closed-loop System with Khin ..................117
8.7 H∞ Loop-shaping Design . . ...............................125
8.8 Assessment of H∞ Loop-shaping Design . . ..................128
8.9 µ-Synthesis and D-K Iterations . . . .........................131
8.10 Robust Stability and Performance of Kmu ..................141
8.11 Comparison of H∞, H∞ LSDP and µ-controllers ............150
8.12 Order Reduction of µ-controller ...........................158
8.13 Conclusions .............................................162Contents xiii
9 A Triple Inverted Pendulum Control-system Design .......163
9.1 System Description . .....................................164
9.2 Modelling of Uncertainties . ...............................167
9.3 Design Specifications .....................................180
9.4 System Interconnections . . ................................182
9.5 H∞ Design . . . ..........................................186
9.6 µ-Synthesis . . . ..........................................191
9.7 Nonlinear System Simulation . . . ..........................199
9.8 Conclusions .............................................202
10 Robust Control of a Hard Disk Drive ......................203
10.1 Hard Disk Drive Servo System ............................203
10.2 Derivation of Uncertainty Model ..........................209
10.3 Closed-loop System-design Specifications . . .................215
10.4 System Interconnections . . ................................218
10.5 Controller Design in Continuous-time ......................219
10.5.1 µ-Design . . .......................................221
10.5.2 H∞ Design .......................................228
10.5.3 H∞ Loop-shaping Design...........................228
10.6 Comparison of Designed Controllers .......................229
10.7 Controller-order Reduction . ..............................237
10.8 Design of Discrete-time Controller . . . ......................239
10.9 Nonlinear System Simulation . . . ..........................244
10.10Conclusions .............................................247
11 Robust Control of a Distillation Column ...................249
11.1 Introduction . ...........................................249
11.2 Dynamic Model of the Distillation Column .................250
11.3 Uncertainty Modelling ...................................254
11.4 Closed-loop System-performance Specifications . . ............256
11.5 Open-loop and Closed-loop System Interconnections .........261
11.6 Controller Design . .......................................261
11.6.1 Loop-shaping Design ...............................262
11.6.2 µ-Synthesis .......................................271
11.7 Nonlinear System Simulation . . . ..........................283
11.8 Conclusions .............................................286
12 Robust Control of a Rocket ................................289
12.1 Rocket Dynamics ........................................289
12.2 Uncertainty Modelling ...................................301
12.3 Performance Requirements . . .............................306
12.4 H∞ Design . . . ..........................................310
12.5 µ-Synthesis . . . ..........................................315
12.6 Discrete-time µ-Synthesis . . ...............................324
12.7 Simulation of the Nonlinear System . .......................328xiv Contents
12.8 Conclusions .............................................332
13 Robust Control of a Flexible-Link Manipulator ............335
13.1 Dynamic Model of the Flexible Manipulator ................336
13.2 A Linear Model of the Uncertain System . . .................339
13.3 System-performance Specifications . ........................355
13.4 System Interconnections . . ................................359
13.5 Controller Design and Analysis . ...........................361
13.6 Nonlinear System Simulations . ............................372
13.7 Conclusions .............................................375
References .....................................................377
Index ..........................................................387
2011-09-10
matlab揭秘 郑碧波译
译言.......................................................................................................................................... I
Mc Graw Hill的介绍............................................................................................................... II
关于作者................................................................................................................................III
目录........................................................................................................................................IV
前言...................................................................................................................................... VII
致谢.....................................................................................................................................VIII
第一章MATLAB环境.................................. 1
用户界面概述...................................................................................................................2
命令窗口与算法基础.......................................................................................................2
赋值运算符.......................................................................................................................4
基本数学定义式...............................................................................................................7
复数..................................................................................................................................8
修正输入...........................................................................................................................9
文件基础...........................................................................................................................9
结束MATLAB ................................................................................................................10
习题................................................................................................................................10
第二章向量与矩阵................................. 11
向量................................................................................................................................12
从已存变量创建大向量.................................................................................................13
创建等差元素向量.........................................................................................................14
特征化向量(Characterizing a Vector) .......................................................................15
向量的点乘和叉乘(数量积和向量积).....................................................................18
引用向量元素.................................................................................................................19
矩阵基本操作.................................................................................................................19
矩阵相乘.........................................................................................................................21
更多基本操作.................................................................................................................22
特殊类型矩阵.................................................................................................................23
引用矩阵元素.................................................................................................................23
行列式与线性系统求解.................................................................................................24
求矩阵的秩.....................................................................................................................25
求逆矩阵与伪逆矩阵.....................................................................................................27
简化梯形矩阵.................................................................................................................30
矩阵分解.........................................................................................................................32
习题................................................................................................................................33
第三章绘图与图形................................. 34
2D绘图基础....................................................................................................................35
更多2D绘图选项...........................................................................................................38
坐标轴命令.....................................................................................................................40
在同一图象中显示多个函数.........................................................................................41
添加图例.........................................................................................................................43
设置颜色.........................................................................................................................44
设置坐标比例.................................................................................................................45
子图................................................................................................................................48
图象重叠和linspace命令...............................................................................................51
极坐标和对数图象.........................................................................................................53
离散数据绘图.................................................................................................................57
等高线图.........................................................................................................................61
三维图象.........................................................................................................................66
习题................................................................................................................................70
第四章统计和MATLAB编程介绍....................... 71
创建柱状图.....................................................................................................................72
基本统计.........................................................................................................................76
编写MATLAB函数........................................................................................................77
使用for循环编程............................................................................................................80
计算标准偏差和中位数.................................................................................................80
更多编程要点.................................................................................................................84
习题................................................................................................................................87
第五章代数方程求解和其它符号工具................. 88
解基本代数方程.............................................................................................................89
二次方程求解.................................................................................................................90
符号方程绘图.................................................................................................................91
高次方程求解.................................................................................................................96
方程组.............................................................................................................................98
方程展开与合并.............................................................................................................99
使用指数和对数函数求解方程...................................................................................101
函数的级数表示...........................................................................................................103
习题...............................................................................................................................105
第六章基本符号演算和微分方程.................... 107
极限计算.......................................................................................................................108
导数计算.......................................................................................................................113
dsolve命令....................................................................................................................118
常微分方程(ODE)求解................................................................................................119
方程组和相平面图.......................................................................................................125
习题...............................................................................................................................132
第七章ODE的数值解............................... 133
使用ODE23 和ODE45 求解一阶方程........................................................................134
二阶方程求解...............................................................................................................140
习题...............................................................................................................................147
第八章积分...................................... 148
INT命令........................................................................................................................149
定积分...........................................................................................................................151
多重积分.......................................................................................................................156
数值积分.......................................................................................................................157
正交积分.......................................................................................................................161
习题...............................................................................................................................162
第九章变换...................................... 163
拉普拉斯变换...............................................................................................................164
拉普拉斯逆变换...........................................................................................................165
微分方程求解...............................................................................................................169
傅立叶变换的计算.......................................................................................................172
傅立叶逆变换...............................................................................................................175
快速傅立叶变换...........................................................................................................175
习题...............................................................................................................................178
第十章曲线拟合.................................. 180
线性函数拟合...............................................................................................................181
指数函数的拟合...........................................................................................................191
习题...............................................................................................................................191
第十一章使用特殊函数工作........................ 193
Γ(伽马)函数..................................................................................................................194
MATLAB中的伽马函数..............................................................................................194
与伽马函数相关的数...................................................................................................197
贝塞耳函数...................................................................................................................198
贝塔函数.......................................................................................................................204
特殊积分.......................................................................................................................205
勒让德函数...................................................................................................................208
亚里函数.......................................................................................................................210
习题...............................................................................................................................212
附录A最终测试................................... 213
附录B习题及测试答案............................. 220
第一章:MATLAB环境..............................................................................................221
第二章:向量与矩阵...................................................................................................221
第三章:绘图与图形...................................................................................................221
第四章:统计和MATLAB编程介绍..........................................................................222
第五章:代数方程求解和其它符号工具...................................................................223
第六章:基本符号演算和微分方程...........................................................................223
第七章:ODE的数值解...............................................................................................223
第八章:积分...............................................................................................................227
第九章:变换...............................................................................................................227
第十章:曲线拟合.......................................................................................................227
第十一章:使用特殊函数工作...................................................................................227
最终测试.......................................................................................................................229
2011-07-27
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