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11 Reliable Computations
11-4
Conditioning and Numerical Stability
Two of the key concepts in numerical analysis are the conditioning of problems
and the stability of algorithms.
Conditioning
Consider the linear system given by
A =
0.7800 0.5630
0.9130 0.6590
b =
0.2170
0.2540
Thetruesolutionisx = [1, –1]' and youcan calculate it approximately using
MATLAB.
x = A\b
x =
1.0000
–1.0000
format long, x
x =
0.99999999991008
–0.99999999987542
Of course, in real problems you almost never have the luxury of knowing the
true solution. This problem is very ill-conditioned. To see this, add a small
perturbation to
A
E =
0.0010 0.0010
–0.0020 –0.0010
and solve the perturbed system
xe = (A+E)\b
xe =
–5.0000
7.3085
Ax b=
AE+
()
xb=
- Control System 1
- How to Contact The MathWorks: 2
- Operations on LTI Models 4
- SISO Design Tool Reference 9
- LTI Viewer Reference 10
- Function Reference 12
- Contents 14
- 1 Introduction 18
- LTI Models 19
- 2 LTI Models 20
- Other Uses of FRD Models 21
- LTI Properties and Methods 22
- Precedence Rules 23
- Command Summary 24
- Creating LTI Models 26
- SISO Transfer Function Models 26
- MIMO Transfer Function Models 28
- Pure Gains 29
- MIMO Zero-Pole-Gain Models 31
- Discrete-Time Models 37
- Data Retrieval 41
- LTI Properties 43
- Direct Property Referencing 49
- Sample Time 51
- Input Names and Output Names 53
- )oranoutputgroup(O) 55
- Model Conversion 58
- Automatic Conversion 59
- Time Delays 61
- Distillation Column Example 63
- Resizing LTI Systems 83
- Arithmetic Operations 85
- Multiplication 87
- Transposition 88
- Pertransposition 88
- Zero-Order Hold 94
- First-Order Hold 96
- Tustin Approximation 96
- Matched Poles and Zeros 97
- Amplitude 99
- 3 Operations on LTI Models 100
- References 101
- Model Analysis Tools 103
- 4 Model Analysis Tools 104
- General Model Characteristics 105
- Model Dynamics 107
- State-Space Realizations 110
- Arrays of LTI Models 113
- 5 Arrays of LTI Models 114
- Introduction 114
- The Concept of an LTI Array 116
- Building LTI Arrays 124
- Indexing Into LTI Arrays 132
- Operations on LTI Arrays 136
- Customization 143
- 6 Customization 144
- User Preferences 145
- SISO Design Tool 145
- LTI Viewer 145
- Setting Toolbox 147
- Preferences 147
- 7 Setting Toolbox Preferences 148
- Units Page 149
- Style Page 149
- SISO Tool Page 151
- Setting Toolbox Preferences 152
- Setting Tool Preferences 153
- 8 Setting Tool Preferences 154
- Parameters Page 157
- Grids Panel 160
- Fonts Panel 160
- Colors Panel 161
- Customizing Response 165
- Plot Properties 165
- Property Editor is 166
- Property Editor 167
- Labels Page 168
- Limits Page 168
- Characteristics Page 172
- Property Editing for Subplots 174
- Auto-Scale checkboxes 177
- Options Page 178
- Design Case Studies 185
- 10 Design Case Studies 186
- Open-Loop Analysis 189
- Root Locus Design 193
- WOF = -K * H; 201
- (2,2) I/O pair 203
- Hard Disk Drive 204
- D = zpk(0.85,0,1,Ts) 209
- K = 4.11e+03; 211
- LQG Design for the x-Axis 218
- LQG Design for the y-Axis 225
- Cross-Coupling Between Axes 227
- Kalman Filtering 234
- Steady-State Design 235
- Now simulate with lsim 239
- Time-Varying Kalman Filter 241
- Reliable Computations 247
- 11 Reliable Computations 248
- Numerical Stability 252
- Choice of LTI Model 254
- Zero-Pole-Gain Models 259
- GUI Reference 265
- 12 GUI Reference 266
- Reference 267
- 13 SISO Design Tool Reference 268
- Menu Bar 270
- System Name 271
- Import From 271
- System Data 271
- Exporting to the Workspace 272
- Exporting to a MAT-file 273
- Print to Figure 275
- Undo and Redo 275
- Root Locus and Bode Diagrams 275
- Customizing Loop Responses 279
- Conversion domain 281
- Draw a Simulink Diagram 283
- Compensator 283
- Changing the gain 285
- Changing the format 285
- Adding poles and zeros 286
- Deleting poles and zeros 286
- Retrieve 287
- Tool Bar 289
- Current Compensator 290
- Feedback Structure 291
- Root Locus Right-Click Menus 292
- Delete Pole/Zero 293
- Edit Compensator 293
- Settling Time 294
- Peak Overshoot 294
- Damping Ratio 294
- Natural Frequency 294
- Zoom opens this submenu 299
- Status Panel 301
- 14 LTI Viewer Reference 304
- LTI Viewer Menu Bar 306
- New Viewer 307
- Import Using the LTI Browser 307
- Setting Preferences 313
- Ordering Properties 313
- Plot Type 315
- Step Response 317
- Impulse Response 318
- Bode Diagram 319
- Bode Magnitude 320
- Nyquist Diagrams 321
- Nichols Charts 322
- , which is the largest 323
- Pole/Zero 324
- Properties 325
- Axis Grouping 327
- I/O Selector 327
- Right-Click Menus for 331
- Response Plots 331
- - regroup the plots 333
- - Select subsets of I/O pairs 333
- Characteristics 334
- Properties online 337
- 16 Function Reference 342
- Functions by Category 343
- Syntax S = allmargin(sys) 352
- 16append 353
- Feedback connection 355
- Parallel connection 355
- Series connection 355
- Syntax asys = augstate(sys) 356
- Syntax sysb = balreal(sys) 357
- Phase (deg); Magnitude (dB) 359
- Syntax bode(sys) 361
- Syntax bodemag(sys) 366
- Syntax sysd = c2d(sys,Ts) 367
- Example Consider the system 368
- Modal Form 370
- Companion Form 370
- Examples Example 1 373
- Example 2 374
- Example w = logspace(1,2,2); 376
- 16connect 377
- Syntax [P,Q] = covar(sys,W) 382
- Syntax Co = ctrb(A,B) 385
- Syntax sysc = d2c(sysd) 389
- Syntax sys1 = d2d(sys,Ts) 392
- Syntax [Wn,Z] = damp(sys) 393
- 1746–1754 397
- 16dcgain 398
- Syntax X = dlyap(A,Q) 402
- Syntax sys = drss(n) 403
- Syntax sys = dss(a,b,c,d,e) 406
- Example The command 406
- 16dssdata 408
- Compute (transmission) zeros 410
- Syntax est = estim(sys,L) 411
- 16evalfr 413
- Example 3 417
- Syntax sys = filt(num,den) 418
- Example Typing the commands 419
- Example Type the commands 421
- Create transfer functions 422
- Create zero-pole-gain models 422
- Set model properties 424
- 16freqresp 425
- 16gensig 428
- Syntax hasdelay(sys) 434
- 16impulse 435
- Syntax impulse(sys) 435
- Syntax initial(sys,x0) 439
- Impulse response 441
- LTIsystemviewer 441
- Step response 441
- 16interp 442
- Purpose Invert LTI systems 443
- Syntax isys = inv(sys) 443
- Example Consider 443
- Syntax boo = isct(sys) 445
- Syntax boo = isempty(sys) 446
- Example Both commands 446
- Syntax boo = isproper(sys) 447
- Syntax boo = issiso(sys) 448
- 16kalman 449
- (Measurement noise) 450
- Syntax sys = lft(sys1,sys2) 455
- Syntax rlqg = lqgreg(kest,k) 457
- Syntax [K,S,e] = lqr(A,B,Q,R) 461
- Syntax lsim(sys,u,t) 465
- Purpose Help on LTI models 470
- Syntax ltimodels 470
- 16ltiview 472
- Syntax ltiview 472
- See Also bode Bode response 473
- Nichols response 474
- Nyquist response 474
- Pole/zero map 474
- Singular value response 474
- 16margin 477
- Syntax sysr = minreal(sys) 480
- Example The commands 480
- Purpose Model order reduction 482
- Syntax n = ndims(sys) 486
- Example sys = rss(3,1,1,3); 486
- Syntax ngrid 487
- Nichols plots 488
- 16nichols 489
- Syntax nichols(sys) 489
- Algorithm See bode 491
- See Also bode Bode plot 491
- 16nyquist 496
- Syntax nyquist(sys) 496
- Syntax Ob = obsv(A,B) 500
- Example Determine if the pair 500
- MATLAB responds with 501
- Syntax [A,B,C,D] = ord2(wn,z) 504
- Syntax [num,den] = pade(T,N) 505
- N>10 should be avoided 506
- 16parallel 508
- Purpose Pole placement design 510
- Syntax K = place(A,B,p) 510
- Syntax pzmap(sys) 513
- Syntax rsys = reg(sys,K,L) 515
- Example sys = rss(4,1,1,2,3); 518
- 16rlocus 519
- Purpose Evans root locus 519
- Syntax rlocus(sys) 519
- See Also pole System poles 521
- Syntax sys = rss(n) 522
- 16series 524
- Property 527
- Syntax sgrid 533
- Plot pole-zero map 534
- Plot root locus 534
- Generate z-plane grid lines 534
- Syntax sigma(sys) 535
- 16sisotool 539
- Syntax sisotool 539
- P to plant and the 540
- Syntax size(sys) 542
- Test if LTI model is empty 543
- Test if LTI model is SISO 543
- Syntax msys = sminreal(sys) 544
- Syntax sys = ss(a,b,c,d) 546
- Conversion to State Space 547
- Syntax sysT = ss2ss(sys,T) 550
- Syntax [sysb,T] = ssbal(sys) 551
- 16ssdata 553
- Syntax step(sys) 555
- Syntax sys = tf(num,den) 558
- Example 4 561
- Discrete-Time 562
- Conventions 562
- 16tfdata 565
- Specify transfer functions 567
- Syntax td = totaldelay(sys) 568
- Syntax zgrid 570
- Generate s-plane grid lines 571
- Syntax sys = zpk(z,p,k) 572
- Variable 574
- Selection 574
- Example Example 1 574
- Syntax [z,p,k] = zpkdata(sys) 577
- Specify zero-pole-gain models 579
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