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the discretized plant has equations
and the weighting matrices for the equivalent discrete cost function are
The integrals are computed using matrix exponential formulas due to Van
Loan (see [2]). The plant is discretized using
c2d and the gain matrix is
computed from the discretized data using
dlqr.
Limitations The discretized problem data should meet the requirements for dlqr.
See Also c2d Discretization of LTI model
dlqr State-feedback LQ regulator for discrete plant
kalmd Discrete Kalman estimator for continuous plant
lqr State-feedback LQ regulator for continuous plant
References [1] Franklin, G.F., J.D.Powell, and M.L.Workman, DigitalControlofDynamic
Systems, Second Edition, Addison-Wesley, 1980, pp. 439–440
[2] Van Loan, C.F., “Computing Integrals Involving the Matrix Exponential,”
IEEE Trans. Automatic Control, AC-15, October 1970.
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- 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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