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Kalman Filtering
9-57
The error covariance before filtering (measurement error) is
MeasErrCov
MeasErrCov =
1.1138
while the error covariance after filtering (estimation error) is only
EstErrCov
EstErrCov =
0.2722
Time-Varying Kalman Filter
The t ime-varying Kalman filter is a gen e ra lization o f the steady-state filter for
time-varying systems or L TI systems w ith nonstationary noise covariance.
Given t he plant state and measurement equations
the time-v arying Kalman filter is g iven by the recursions
Measurement update
Time update
xn 1+[]Ax n[] Bu n[] Gw n[]++=
y
v
n[] Cx n[] vn[]+=
x
ˆ
nn[]x
ˆ
nn 1–[]Mn[]y
v
n[] Cx
ˆ
nn 1–[]–()+=
Mn[] Pnn 1–[]C
T
Rn[] CP n n 1–[]C
T
+()
1–
=
Pnn[]IMn[]C–()Pnn 1–[]=
x
ˆ
n1n+[]Ax
ˆ
nn[]Bu n[]+=
Pn 1n+[]AP n n[]A
T
GQ n[]G
T
+=
- 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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