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Identifying Input-Output Polynomial Models
For a system with nu inp uts and ny outputs, A(q)isanny-by- ny matrix. A(q)
can be represented as a polynomial in the shift operator q
-1
:
Aq I Aq A q
ny na
na
()=+ ++
−−
1
1
…
For more information about the time-shift operator, see “Understanding the
Time-Shift Operator q” on page 3-43.
A(q) can also be represented as a matrix:
Aq
aq aq a q
aq aq a q
aqa
ny
ny
ny n
()
() () ()
() () ()
()
=
11 12 1
21 22 2
1
…
…
…………
yynynyq
qa
2
()
()
…
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
where the matrix element a
kj
is a polynomial in the shift operator q
-1
:
aq aq a q
kj kj kj
kj
na
na
kj
kj
()=+ ++
−
−
δ
11
…
δ
kj
represents the Kronecker delta, w hich equals 1 for k=j and equals 0
for k≠j. This polynomial describes h ow the old values of the jth output are
affected by the kth output. The ith row of A(q) represents the contribution of
the past output values for predict the current value of the ith output.
B(q)isanny-by-ny matrix. B(q) can be represented as a polynomial in the
shift operator q
-1
:
Bq B Bq B q
nb
nb
()=+ ++
−−
01
1
…
3-47
- User’s Guide 1
- 508-647-7000 (Phone) 2
- 508-647-7001 (Fax) 2
- The MathWorks, Inc 2
- 3 Apple Hill Drive 2
- Natick, MA 01760-2098 2
- Revision History 3
- About the Developers 5
- Linear Model Identification 10
- Modeling) 15
- Model Analysis 17
- Simulation and Prediction 19
- Contents 22
- Types of Data You Can Model 27
- Sampling Intervals 28
- - For a data set with N 29
- SamplingInstants 30
- Ys GsUs() () ()= 33
- Representing Data in the GUI 35
- Importing Time 37
- Domain Data into the GUI 37
- 0. For more information 41
- Prerequisite 43
- Specifying 51
- Creating Multie 55
- Extracting 59
- Set into a New Data Set 59
- Viewing Data Properties 62
- Information About the Data 64
- Distinguishing 65
- Data Types i n the GUI 65
- Organizing Data Icons 65
- Subreferencing iddata Objects 77
- Naming, Adding 85
- Concatenating i 87
- Concatenating idfrd Objects 93
- Supported Data Plots 97
- How to Plot Data in the GUI 98
- Working with a Time Plot 99
- 1 Data Processing 100
- Selecting Subsets of Data 108
- Selecting Data U 109
- Handling Outliers 113
- See Also 115
- When Not to Detrend Data 117
- For example, Ynew = Ysim+y0 121
- Resampling Data 122
- Using the GUI 123
- Resampling 123
- Data at the Command Line 123
- Filtering D ata 129
- Filtering Data 131
- Simple Passband Filter 133
- Causal and Noncausal Filters 135
- Transforming Time-Domain Data 141
- Data Processing 146
- Transform 149
- Choosing Your System 155
- Identification Strategy 155
- Continuous-Time Models 158
- Discrete-Time Models 158
- ODEs (Grey-Box Models) 158
- Nonlinear Models 159
- Nonlinear Black-Box Models 160
- Commands for Model Estimation 163
- Model Proper ties 168
- - Algorithm 169
- - EstimationIn fo 170
- 3 Linear Model Identification 180
- Yz GzUz() () ()= 180
- Data Suppor ted b 181
- How to Estimate 181
- How to Estimate F 183
- Command Line 183
- What Is Frequency Resolution? 185
- Spectral Analysis Algorithm 186
- Linear Model Identification 188
- () ( )=− 193
- Step Respo nse Plot 197
- Gqut gkut k 198
- ()() ()( )=− 198
- What Is a Process Model? 200
- All real or 203
- Underdamped 203
- Options for Initial States 217
- Polynomial Model Structure 220
- Ty pes of Supported Data 226
- Input Delays 227
- [1:10 1:10 1:10] 229
- 2 and that there is a good 233
- Number of parameters =+nn 234
- Before You Begin 235
- None. Skipping uncertainty 237
- Aqyt B qu t nk Cqet 245
- ()() () ()()=− 245
- For example: 249
- What Are State-Space Models? 251
- Discrete-Time Representation 253
- State Matrices 253
- Data Supported b 255
- Why Estimate Model Orders? 257
- State Space: n [n k] 263
- 4[11]for a fourth-order 263
- Supported State-Space Models 265
- Parameterization 269
- Structured Parameterization? 274
- When to Refine Models 281
- Guesses at the Command Line 285
- Representations 290
- Effects on the Noise Model 293
- Subreferencing Model Objects 297
- Concatenating Model Objects 302
- Merging Model Objects 306
- Nonlinear Black-Box Model 307
- Identification 307
- and linear trends 308
- Using Regressors 312
- 1 because the prediction of 313
- 2), y ou 313
- Using Custom Regressors 314
- None for t he Nonlinearity 315
- How to Estimate N 316
- General nlarx Syntax 318
- Ha mmerstein-Wiener 324
- TypesofNonlinearityEstimators 331
- Hammerstein-Wi 337
- Nonlinear M odel? 339
- Models for a Given Input 340
- ODE Parameter Estimation 345
- (Grey-Box Modeling) 345
- Supported Grey-Box Models 346
- Model for Heat Diffusion 353
- Model = pem(data,Minit) 359
- Simulation Method 364
- Search Method 364
- Gradient Options 365
- Time Series Model 369
- What Are Time-Series Models? 370
- Preparing Time-Series Data 371
- Linear or Logarithmic 372
- Linear option is available 372
- Estimating AR and ARMA Models 375
- ARMA:[na nc] 376
- Ny-by-Ny 376
- Recursive Techniques for 385
- Model Identi fication 385
- What Is Recursive Estimation? 386
- Kalman Filter Algorithm 392
- Forg etting Factor Algorithm 394
- 0.97 and 0.995 395
- 0.9 7 to 0.995 395
- Algorithm 396
- Data Segmentation 398
- 8 Model Analysis 400
- Plotting Models in the GUI 403
- Active model 404
- Inactive model 404
- Plots the model 404
- Getting Advice A 405
- Supported Model Types 415
- How to Plot Resid 419
- Example – Ex 419
- OutputName is the name of the 427
- 0 and the 427
- What Is Frequency Response? 430
- How to Plo 433
- [1:.1:100] 435
- Creating Noise-Spectrum Plots 438
- Supported Models 445
- Reducing Model O 449
- Definition of FPE 459
- Computing FPE 460
- Definition of AIC 460
- Computing AIC 461
- Computing Model Uncertainty 462
- Troubleshooting Models 465
- Substantial Noi 467
- Unstable M 467
- Complicated Nonlinearities 469
- 9 Simulation and Prediction 474
- Model at the Comm 478
- 479
- Specifying Initial States 480
- Using Identified Models in 483
- Control Design 483
- Identified Models 484
- Using balred to R 485
- Compensator D 485
- Software 485
- What Is the LTI Viewer? 487
- Combining Model Objects 488
- Using System Identification 491
- Toolbox Blocks 491
- Preparing Data 494
- Identifying L inear Models 495
- Simulating Model Output 496
- Using the System 501
- Identification Tool GUI 501
- Starting a New Se 504
- Tool GUI 505
- Opening a Saved Session 506
- Deleting a Session 507
- Getting Help in the GUI 507
- Managing Models in the GUI 509
- Viewing Model Pr 510
- Renaming Models 511
- Organizing 511
- Model Icons 511
- Deleting Models in the GUI 512
- Workspace 513
- Changing 516
- Setting Axis Limits 517
- Grid, Line Style 519
- Opening a Plo 519
- Printing Plots 520
- TypesofGUICustomization 521
- Saving Session Preferences 521
- Modifying idlayout.m 522
- levels 1-94 529
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