MATLAB CONTROL SYSTEM TOOLBOX 9 Guida Utente Pagina 481
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gram
11-91
11gram
Purpose Compute controllability and observability gramians
Syntax Wc = gram(sys,'c')
Wo = gram(sys,'o')
Description gram calculates controllability and observability gramians. You can use
gramians to study the controllability and observability properties of
state-space models and for model reduction [1 ,2]. They have better numerical
properties than the controllability and observability matrices formed by
ctrb
and obsv.
Given t he continuous-time state-space model
the controllability gramian is defined by
and the observability gramian by
The discrete-time counterparts are
The controllability gramian is positive definite if and only if is
controllable. Similarly, the observability gramian is positive definite if and
only if is observable.
Use the commands
Wc = gram(sys,'c') % controllability gramian
Wo = gram(sys,'o') % observability gramian
x
·
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e
Aτ
BB
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T
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T
Ce
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0
∞
∫
=
W
c
A
k
BB
T
A
T
()
k
,
k0=
∞
∑
= W
o
A
T
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k
C
T
CA
k
k 0=
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=
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- Control System 1
- How to Contact The MathWorks: 2
- Control Design Tools 10
- The Root Locus Design GUI 11
- Design Case Studies 12
- Reliable Computations 13
- Reference 13
- Contents 14
- Preface 16
- Installation 17
- Typographic Conventions 19
- Quick Start 21
- 1 Quick Start 22
- LTI Models 23
- Model Conversion 25
- LTI Properties 27
- System Interconnections 39
- Other Uses of FRD Models 47
- LTI Objects 47
- Precedence Rules 49
- Creating LTI Models 53
- Pure Gains 55
- MIMO Zero-Pole-Gain Models 57
- Direct Property Referencing 77
- Automatic Conversion 87
- Time Delays 89
- Distillation Column Example 91
- 2 LTI Models 100
- References 103
- Operations on LTI Models 105
- 3 Operations on LTI Models 106
- Resizing LTI Systems 113
- Arithmetic Operations 115
- Multiplication 117
- Concatenation of LTI Models 121
- Matched Poles and Zeros 127
- Arrays of LTI Models 133
- 4 Arrays of LTI Models 134
- Introduction 135
- The Concept of an LTI Array 137
- Building LTI Arrays 145
- Indexing Into LTI Arrays 153
- Operations on LTI Arrays 157
- Dimension Requirements 159
- Model Analysis Tools 163
- 5 Model Analysis Tools 164
- General Model Characteristics 165
- Model Dynamics 167
- State-Space Realizations 169
- Time and Frequency Response 171
- Frequency Response 173
- Customizing the Plot Display 179
- The LTI Viewer 183
- 6 The LTI Viewer 184
- File menu has 189
- Tools menu items 189
- 1 Click on the marker 191
- Importing Models 193
- The LTI Viewer Menus 197
- Interactive Help 199
- The Right-Click Menus 201
- Characteristics or the 203
- The Axes Grouping Submenu 205
- Axes Grouping 205
- Shift key while 209
- Arrays pull-down tab 213
- Index into Dimensions 217
- The LTI Viewer Tools Menu 221
- 2 Select either: 229
- Simulink LTI Viewer 231
- A Sample Analysis Task 231
- Model_Inputs_and_Outputs 233
- Get Line arized Mod el 239
- Linearized Model 239
- Performing Linear Analysis 243
- Saving Analysis Models 247
- 7 Control Design Tools 250
- Root Locus Design 251
- Pole Placement 253
- Optimal State-Feedback Gain 257
- Kalman State Estimator 257
- LQG Design 259
- The Root Locus Design 261
- 8 The Root Locus Design GUI 262
- A Servomechanism Example 265
- Specifying the Design Model 271
- Gain text boxon the 273
- As you get close to a closed 275
- Zoom button to zoom in 277
- Gain,andpressingthe 281
- Gain box 283
- Add Grid/Boundary menu 285
- Tools menu 285
- Edit Compensator menu item to 287
- Add compensator poles here 289
- Compensator 293
- Marker for the 295
- Then type 297
- Designating the Model Source 299
- Listing Poles and Zeros 301
- Show Ob ject buttons in this 303
- OK to close this window 303
- The plant, P (Gservo) 305
- The root locus 305
- Sensor dynamics, H 305
- Pre-filter, F 305
- 9 Design Case Studies 310
- H = zpk(0,–0.333,1); 323
- D = zpk(0.85,0,1,Ts) 333
- LQG Regulation 339
- Cross-Coupling Between Axes 351
- MIMO LQG Design 355
- Steady-State Design 359
- Kalman Filtering 361
- Time-Varying Kalman Filter 365
- 10 Reliable Computations 374
- Choice of LTI Model 381
- 11 Reference 392
- Category Tables 393
- 11append 402
- Example The commands 403
- Feedback connection 404
- Parallel connection 404
- Series connection 404
- Syntax asys = augstate(sys) 405
- Syntax sysb = balreal(sys) 406
- Phase (deg); Magnitude (dB) 407
- Algorithm Consider the model 408
- Syntax bode(sys) 409
- Syntax sysd = c2d(sys,Ts) 414
- Example Consider the system 414
- Modal Form 417
- Companion Form 417
- Examples Example 1 420
- Example 2 421
- Example w = logspace(1,2,2); 423
- 11connect 424
- Center, 1976 429
- Syntax [P,Q] = covar(sys,W) 430
- Syntax Co = ctrb(A,B) 433
- Syntax sysc = d2c(sysd) 438
- Syntax sys1 = d2d(sys,Ts) 441
- Syntax [Wn,Z] = damp(sys) 442
- 1746–1754 446
- 11dcgain 447
- Syntax sys = delay2z(sys) 448
- Syntax X = dlyap(A,Q) 451
- Syntax sys = drss(n) 452
- Syntax sys = dss(a,b,c,d,e) 455
- Example The command 456
- 11dssdata 457
- Syntax s = esort(p) 458
- , sort Sort system poles 459
- Compute system poles 459
- Pole-zero map 459
- Compute (transmission) zeros 459
- Syntax est = estim(sys,L) 460
- 11evalfr 462
- 11feedback 463
- Example 3 466
- Syntax sys = filt(num,den) 467
- Example Typing the commands 468
- Example Type the commands 470
- Create transfer functions 471
- Create zero-pole-gain models 471
- Set model properties 473
- 11freqresp 474
- 11gensig 477
- Plot the resulting signal 478
- Syntax hasdelay(sys) 483
- Syntax impulse(sys) 484
- Amplitude 486
- Syntax initial(sys,x0) 488
- Purpose Invert LTI systems 491
- Syntax isys = inv(sys) 491
- Example Consider 491
- Syntax boo = isct(sys) 494
- Syntax boo = isempty(sys) 495
- Example Both commands 495
- Syntax boo = isproper(sys) 496
- Syntax boo = issiso(sys) 497
- Continuous-Time Estimation 498
- Discrete-Time Estimation 499
- Syntax sys = lft(sys1,sys2) 505
- Syntax rlqg = lqgreg(kest,k) 507
- Syntax [K,S,e] = lqr(A,B,Q,R) 511
- Form LQG regulator 512
- Syntax lsim(sys,u,t) 516
- Then simulate with lsim 518
- Syntax ltiview 523
- See Also bode Bode response 524
- 11margin 527
- MATLAB responds with 528
- Syntax sysr = minreal(sys) 530
- M odel order reduction 531
- Structured model reduction 531
- Purpose Model order reduction 532
- Syntax n = ndims(sys) 536
- Example sys = rss(3,1,1,3); 536
- Syntax ngrid 537
- Open−Loop Gain (dB) 538
- Syntax nichols(sys) 539
- See Also bode Bode plot 545
- Syntax nyquist(sys) 546
- Syntax Ob = obsv(A,B) 549
- Example Determine if the pair 549
- Syntax [A,B,C,D] = ord2(wn,z) 553
- Syntax [num,den] = pade(T,N) 554
- N>10 should be avoided 556
- 11parallel 558
- Purpose Pole placement design 560
- Syntax K = place(A,B,p) 560
- Syntax pzmap(sys) 563
- Imag Axis 564
- Syntax rsys = reg(sys,K,L) 566
- Example sys = rss(4,1,1,2,3); 569
- 11rlocfind 570
- Purpose Evans root locus 572
- Syntax rlocus(sys) 572
- Syntax rltool 575
- F and H 578
- Syntax sys = rss(n) 579
- 11series 581
- Property 584
- Syntax sgrid 590
- Syntax sigma(sys) 592
- Nichols plot 596
- Nyquist plot 596
- Syntax size(sys) 597
- Test if LTI model is empty 598
- Test if LTI model is SISO 598
- Syntax msys = sminreal(sys) 599
- Syntax sys = ss(a,b,c,d) 601
- Conversion to State Space 602
- Syntax sysT = ss2ss(sys,T) 605
- Syntax [sysb,T] = ssbal(sys) 606
- 11ssdata 608
- Syntax step(sys) 610
- Syntax sys = tf(num,den) 614
- Example 4 618
- Discrete-Time 618
- Conventions 618
- 11tfdata 621
- Specify transfer functions 623
- Syntax td = totaldelay(sys) 624
- Syntax zgrid 626
- Syntax sys = zpk(z,p,k) 628
- Variable 630
- Selection 630
- Example Example 1 631
- Syntax [z,p,k] = zpkdata(sys) 633
- Get properties of LTI models 635
- Specify zero-pole-gain models 635
- See state-space models 638
- See also LTI models 639
- See transposition 639
- See Kalman estimator 639
- See time response 639
- See I/O groups 640
- See also InputName 640
- See also Simulink LTI Viewer 641
- See delays 641
- See model building 641
- See delay 641
- See frequency response 642
- See also operations 642
- See LTI properties 642
- See Simulink LTI Viewer 643
- See LTI objects 644
- See also OutputName 644
- See transfer functions 647
- See zeros 648
- See zero 648
- See also noise 648
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