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3 Linear Model Identification
The exact relations hip between K and
K
is com plicated. However, for short
sampling intervals T, the following approximation works w el l:
KeKd
F
T
=
∫
τ
τ
0
State-Space Representation of Transfer Functions
For line ar models, the general symbolic model des cription is given by:
yGuHe=+
G is a transfer function that takes the input u to the output y. H is a transfer
function that describes the properties of the additive o u tput noise model.
The discrete-time state-space representation is given by the following
equation:
x kT T Ax kT Bu kT Ke kT
ykT CxkT DukT ekT
x
( ) () () ()
() () () ()
()
+= + +
=+ +
=0 xx0
where T is the sampling interval, u(kT) is the input at time instant kT,and
y(kT) is the output at time instant kT.
The relationships between the transfer functions and the discrete-time
state-space matrices are given by the following equations:
Gq CqI A B D
Hq CqI A K I
nx
nx ny
() ( )
() ( )
=−+
=−+
−
−
1
1
where I
nx
is the nx-by-nx identity matrix, I
ny
is the nx-by-nx identity matrix,
and ny is the dimension of y and e.
3-76
- 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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