Handbook of Model Predictive Control (eBook)

Preface 7
Contents 9
Part I Theory 22
The Essentials of Model Predictive Control 23
1 Introduction 23
2 Background 24
2.1 Intuition 24
2.2 History 25
3 Basics of Model Predictive Control (MPC) 32
4 Stability of MPC 35
5 Exogenous Inputs 37
6 Robustness 39
7 Example 40
7.1 Background 41
7.2 Dynamics 41
7.3 Delay 43
7.4 Performance Measure 43
7.5 Noise and Other Disturbances 44
7.6 Problem 44
7.7 Solution 44
7.8 Results 45
7.9 Discussion 46
8 Conclusions 46
References 46
Dynamic Programming, Optimal Control and Model PredictiveControl 48
1 Introduction 48
2 Setting, Definitions and Notation 49
3 Dynamic Programming 52
4 Stabilizing MPC 54
4.1 Terminal Conditions 55
4.2 No Terminal Conditions 56
5 Economic MPC 59
5.1 Terminal Conditions 60
5.2 No Terminal Conditions 62
6 Conclusions 70
References 70
Set-Valued and Lyapunov Methods for MPC 72
1 Introduction 72
2 Problem Statement and Assumptions 73
2.1 Open Loop Optimal Control Problem 73
2.2 Closed Loop Dynamics 75
2.3 Standing Assumptions 75
3 Properties of the Open Loop Optimal Control Problem 76
3.1 Set-Valued Analysis Background 76
3.2 Parametric Optimization Background 77
3.3 Existence and Structure of Optimal Solutions 79
4 Asymptotic Stability and Related Issues 81
4.1 Strong Positive Invariance (a.k.a. RecursiveFeasibility) 82
4.2 Strong Lyapunov Decrease (a.k.a. Cost Reduction) 83
4.3 Strong Positive Invariance and Strong Asymptotic Stability 84
4.4 Set-Valued Approach to Robustness of Asymptotic Stability 85
4.5 Consistent Improvement 86
5 Set-Valued Control Systems 87
5.1 Weak Formulation of MPC 88
5.2 Strong Formulation of MPC 90
References 91
Stochastic Model Predictive Control 93
1 Introduction 93
2 Stochastic Optimal Control and MPC with Chance Constraints 94
3 Scenario Tree-Based MPC 96
3.1 Scenario-Tree Construction 97
3.2 Scenario-Tree Stochastic Optimization Problem 99
3.3 Extensions and Applications 100
4 Polynomial Chaos-Based MPC 102
4.1 System Model, Constraints, and Control Input Parameterization 102
4.2 Generalized Polynomial Chaos for Uncertainty Propagation 103
4.3 Moment-Based Surrogate for Joint Chance Constraint 106
4.4 Sample-Free, Moment-Based SMPC Formulation 107
4.5 Extensions 108
5 Stochastic Tube MPC 108
5.1 System Model, Disturbance Model and Constraints 108
5.2 Tube MPC Design 109
5.3 Theoretical Guarantees 111
5.4 Mass-Spring-Damper Example 112
5.5 Extensions 112
References 113
Moving Horizon Estimation 116
1 Introduction 116
2 Systems of Interest 120
3 MHE Setup 123
4 Main Results 127
5 Numerical Example 130
6 Conclusions 133
References 139
Probing and Duality in Stochastic Model Predictive Control 142
1 Introduction 142
2 Stochastic Optimal Control and Duality 143
2.1 The State, the Information State, and the BayesianFilter 143
2.2 Stochastic Optimal Control and the Information State 144
2.3 Duality and the Source of Intractability 145
3 Stochastic MPC and Deterministic MPC 145
4 Stochastic Reconstructibility and Its Dependence on Control 146
4.1 Linear Regression and the Cramér-Rao Lower Bound 147
4.2 Conditional Entropy Measure of Reconstructibility 148
5 Three Examples of Dualized Stochastic Control 150
5.1 Internet Congestion Control in TCP/IP 150
5.2 Equalization in Cellular Wireless 151
5.3 Experiment Design in Linear Regression for MPC 154
6 Tractable Compromise Dualized Stochastic MPC Algorithms 156
6.1 Non-dual Approaches 157
6.1.1 Particle-Based Methods 157
6.1.2 Certainty Equivalence Methods 158
6.2 Dual Optimal POMDPs 158
7 Conclusion 159
References 160
Economic Model Predictive Control: Some Design Tools and Analysis Techniques 162
1 Model-Based Control and Optimization 162
2 Formulation of Economic Model Predictive Control 165
3 Properties of Economic MPC 168
3.1 Recursive Feasibility 168
3.1.1 Terminal Equality Constraints 168
3.1.2 Terminal Set (or Terminal Inequality Constraint) 169
3.2 Asymptotic Average Cost 170
3.2.1 Terminal Feasible Trajectory 170
3.2.2 Terminal Penalty Function 171
3.2.3 Adaptive Terminal Weight 172
3.3 Stability of Economic MPC 173
3.4 EMPC Without Terminal Ingredients 177
4 EMPC with Constraints on Average 178
5 Robust Economic Model Predictive Control 179
6 Conclusions 181
References 182
Nonlinear Predictive Control for Trajectory Tracking and Path Following: An Introduction and Perspective 185
1 Introduction and Motivation 186
2 Setpoint Stabilization, Trajectory Tracking, Path Following, and Economic Objectives 189
2.1 Setpoint Stabilization 189
2.2 Trajectory Tracking 190
2.3 Path Following 191
2.4 Economic Objectives 193
3 A Brief Review of MPC for Setpoint Stabilization 193
3.1 Comments on Convergence and Stability 195
3.2 Setpoint Stabilization of a Lightweight Robot 196
4 Model Predictive Control for Trajectory Tracking 197
4.1 Convergence and Stability of Tracking NMPC 198
4.2 Trajectory-Tracking Control of a Lightweight Robot 199
5 Model Predictive Control for Path Following 199
5.1 Convergence and Stability of Output Path-Following NMPC 201
5.2 Path-Following Control of a Lightweight Robot 202
5.2.1 Nominal Case 203
5.2.2 Disturbance Case 205
5.3 Extensions of Path Following 207
6 Economic MPC 208
6.1 Convergence and Stability of Economic MPC 209
7 Conclusions and Perspectives 210
References 211
Hybrid Model Predictive Control 215
1 Summary 215
2 Hybrid Model Predictive Control 216
2.1 Discrete-Time MPC for Discrete-Time Systems with Discontinuous Right-Hand Sides 217
2.2 Discrete-Time MPC for Discrete-Time Systems with Mixed States 219
2.3 Discrete-Time MPC for Discrete-Time Systems Using Memory and Logic Variables 220
2.4 Periodic Continuous-Discrete MPC for Continuous-Time Systems 224
2.4.1 With Piecewise Continuous Inputs 224
2.4.2 With Piecewise Constant Inputs 226
2.5 Periodic Continuous-Time MPC for Continuous-Time Systems Combined with Local Static State-Feedback Controllers 227
2.6 Periodic Discrete-Time MPC for Continuous-Time Linear Systems with Impulses 228
3 Towards MPC for Hybrid Dynamical Systems 231
4 Further Reading 234
References 234
Model Predictive Control of Polynomial Systems 237
1 Introduction 237
2 Model Predictive Control of Discrete-Time Polynomial Systems 238
3 Polynomial Optimization Methods 240
3.1 Sum-of-Squares Decomposition 241
3.2 Dual Approach via SOS Decomposition 241
4 Fast Solution Methods for Polynomial MPC 243
4.1 Convex MPC for a Subclass of Polynomial Systems 243
4.2 Explicit MPC Using Algebraic Geometry Methods 244
5 Taylor Series Approximations for Non-polynomial Systems 246
5.1 Taylor's Theorem 246
5.2 Example 247
6 Outlook for Future Research 249
References 251
Distributed MPC for Large-Scale Systems 254
1 Introduction and Motivations 254
2 Model and Control Problem Decomposition 256
2.1 Model Decomposition 256
2.1.1 Non-overlapping Decompositions 258
2.1.2 Overlapping Decompositions 259
2.2 Partition Properties and Control 259
2.3 MPC Problem Separability 260
3 Decentralized MPC 262
4 Distributed MPC 263
4.1 Cooperating DMPC 263
4.2 Non-cooperating Robustness-Based DMPC 265
4.3 Distributed Control of Independent Systems 267
4.4 Distributed Optimization 268
5 Extensions and Applications 270
6 Conclusions and Future Perspectives 271
References 271
Scalable MPC Design 274
1 Introduction and Motivations 274
2 Scalable and Plug-and-Play Design 275
3 Concepts Enabling Scalable Design for Constrained Systems 278
3.1 Tube-Based Small-Gain Conditions for Networks 278
3.1.1 Tube-Based Small-Gain Condition for Networks Using RPI Sets 280
3.1.2 Tube-Based Small-Gain Condition for Networks Using RCI Sets 281
3.2 Distributed Invariance 281
4 Scalable Design of MPC 283
4.1 PnP-MPC Based on Robustness Against Coupling 283
4.1.1 PnP-MPC Exploiting the Small-Gain Conditions for Networks Using RPI Sets 285
4.1.2 PnP-MPC Exploiting the Small-Gain Conditions for Networks Using RCI Sets 286
4.2 PnP-MPC Based on Distributed Invariance 286
4.2.1 Implementation of Distributed MPC 286
4.2.2 Distributed Synthesis 287
4.2.3 Plug-and-Play and Hot-Transitions 288
5 Generalizations and Related Approaches 289
6 Applications 291
6.1 Frequency Control in Power Networks 291
6.2 Electric Vehicle Charging in Smart Grids 293
7 Conclusions and Perspectives 295
References 296
Part II Computations 299
Efficient Convex Optimization for Linear MPC 300
1 Introduction 300
2 Formulating and Solving LQR 301
3 Convex Quadratic Programming 302
4 Linear MPC Formulations and Interior-Point Implementation 305
4.1 Linear MPC Formulations 305
4.2 KKT Conditions and Efficient Interior-Point Implementation 307
5 Parametrized Convex Quadratic Programming 310
5.1 Enumeration 311
5.2 Active-Set Strategy 312
6 Software 315
References 315
Implicit Non-convex Model Predictive Control 317
1 Introduction 317
2 Parametric Nonlinear Programming 319
3 Solution Approaches to Nonlinear Programming 320
3.1 SQP 321
3.2 Interior-Point Methods 322
4 Discretization 323
4.1 Single Shooting Methods 324
4.2 Multiple Shooting Methods 325
4.3 Direct Collocation Methods 326
5 Predictors & Path-Following
5.1 Parametric Embedding 329
5.2 Path Following Methods 331
5.3 Real-Time Dilemma: Should We Convergethe Solutions? 333
5.4 Shifting 335
5.5 Convergence of Path-Following Methods 336
6 Sensitivities & Hessian Approximation
7 Structures 339
8 Summary 341
References 342
Convexification and Real-Time Optimization for MPC with Aerospace Applications 346
1 Introduction 346
2 Convexification 348
2.1 Lossless Convexification of Control Constraints 349
2.1.1 Theory 350
2.1.2 Application 354
2.2 Successive Convexification 356
2.2.1 Theory 357
2.2.2 Application 361
3 Real-Time Computation 363
4 Concluding Remarks 366
References 367
Explicit (Offline) Optimization for MPC 370
1 Introduction 370
1.1 From State-Space Models to Multi-Parametric Programming 370
1.2 When Discrete Elements Occur 374
2 Multi-Parametric Linear and Quadratic Programming: An Overview 374
2.1 Theoretical Properties 375
2.1.1 Literature Review 378
2.2 Degeneracy 378
2.2.1 Literature Review 379
2.3 Solution Algorithms for mp-LP and mp-QP Problems 380
3 Multi-Parametric Mixed-Integer Linear and Quadratic Programming: An Overview 384
3.1 Theoretical Properties 384
3.1.1 On the Notion of Exactness 385
3.2 Solution Algorithms 386
3.2.1 Literature Overview 386
3.3 The Decomposition Algorithm 388
3.3.1 Calculation of a New Candidate Integer Solution 388
3.3.2 mp-QP Solution 388
3.3.3 Comparison Procedure 389
4 Discussion and Concluding Remarks 390
4.1 Size of Multi-Parametric Programming Problem and Offline Computational Effort 390
4.2 Size of the Solution and Online Computational Effort 391
4.3 Other Developments in Explicit MPC 392
References 393
Real-Time Implementation of Explicit Model Predictive Control 397
1 Simplification of MPC Feedback Laws 397
1.1 Preliminaries 397
1.2 Complexity of Explicit MPC 399
1.3 Problem Statement and Main Results 400
2 Piecewise Affine Explicit MPC Controllers of ReducedComplexity 401
2.1 Clipping-Based Explicit MPC 401
2.2 Regionless Explicit MPC 404
2.3 Piecewise Affine Approximation of Explicit MPC 407
3 Approximation of MPC Feedback Laws for Nonlinear Systems 410
3.1 Problem Setup 410
3.2 A QP-Based MPC Controller 411
3.3 Stability Verification 412
3.3.1 Lypunov Analysis 413
3.3.2 Sum-of-Squares Certificates 414
3.4 Closed-Loop Performance 415
3.5 Parameter Tuning 416
3.5.1 First Phase: Minimization of the SOS Slack 416
3.5.2 Second Phase: Minimization of the Performance Metric 418
3.6 Numerical Example 418
References 420
Robust Optimization for MPC 423
1 Introduction 423
2 Problem Formulation 424
2.1 Inf-Sup Feedback Model Predictive Control 425
2.2 Set-Based Robust Model Predictive Control 426
2.3 Numerical Challenges 428
3 Convex Approximations for Robust MPC 428
3.1 Ellipsoidal Approximation Using LMIs 429
3.2 Affine Disturbance Feedback 431
4 Generic Methods for Robust MPC 433
4.1 Inf-Sup Dynamic Programming 434
4.2 Scenario-Tree MPC 436
4.3 Tube MPC 437
5 Numerical Methods for Tube MPC 438
5.1 Feedback Parametrization 438
5.2 Affine Set-Parametrizations 439
5.3 Tube MPC Parametrization 441
5.4 Tube MPC Via Min-Max Differential Inequalities 441
6 Numerical Aspects: Modern Set-Valued Computing 443
6.1 Factorable Functions 443
6.2 Set Arithmetics 445
6.3 Set-Valued Integrators 447
7 Conclusions 449
References 450
Scenario Optimization for MPC 454
1 Introduction 454
2 Stochastic MPC and the Use of the Scenario Approach 455
3 Fundamentals of Scenario Optimization 457
4 The Scenario Approach for Solving Stochastic MPC 460
5 Numerical Example 465
6 Extensions and Future Work 469
References 470
Nonlinear Programming Formulations for Nonlinear and Economic Model Predictive Control 473
1 Introduction 473
1.1 NLP Strategies for NMPC 474
2 Properties of the NLP Subproblem 475
2.1 NMPC Problem Reformulation 477
3 Nominal and ISS Stability of NMPC 478
4 Economic NMPC with Objective Regularization 480
4.1 Regularization of Non-convex Economic Stage Costs 482
4.2 Economic NMPC with Regularization of ReducedStates 483
5 Economic MPC with a Stabilizing Constraint 489
6 Case Studies 490
6.1 Nonlinear CSTR 490
6.2 Large-Scale Distillation System 492
7 Conclusions 495
References 495
Part III Applications 498
Automotive Applications of Model Predictive Control 499
1 Model Predictive Control in Automotive Applications 499
1.1 A Brief History 500
1.2 Opportunities and Challenges 501
1.3 Chapter Overview 504
2 MPC for Powertrain Control, Vehicle Dynamics, and Energy Management 504
2.1 Powertrain Control 504
2.1.1 MPC Opportunities in Powertrain Control 508
2.2 Control of Vehicle Dynamics 510
2.2.1 MPC Opportunities in Vehicle Dynamics 513
2.3 Energy Management in Hybrid Vehicles 514
2.3.1 MPC Opportunities in Hybrid Vehicles 516
2.4 Other Applications 517
3 MPC Design Process in Automotive Applications 517
3.1 Prediction Model 518
3.2 Horizon and Constraints 521
3.3 Cost Function, Terminal Set and Soft Constraints 522
4 Computations and Numerical Algorithms 524
4.1 Explicit MPC 525
4.2 Online MPC 527
4.3 Nonlinear MPC 528
5 Conclusions and Future Perspectives 529
References 529
Applications of MPC in the Area of Health Care 534
1 Introduction 534
2 Is MPC Relevant to Health Problems? 535
3 Special Characteristics of Control Problems in the Areaof Health 535
3.1 Safety 536
3.2 Background Knowledge 536
3.3 Models 536
3.4 Population Versus Personalised Models 537
4 Specific Examples Where MPC Has Been Used in the Area of Health 537
4.1 Ambulance Scheduling 537
4.2 Joint Movement 539
4.3 Type 1 Diabetes Treatment 540
4.4 Anaesthesia 542
4.5 HIV 543
4.6 Cancer 545
4.7 Inflammation 547
5 Appraisal 548
6 Conclusion 549
References 550
Model Predictive Control for Power Electronics Applications 556
1 Introduction 556
2 Basic Concepts 558
2.1 System Constraints 558
2.2 Cost Function 559
2.3 Moving Horizon Optimization 561
2.4 Design Parameters 562
3 Linear Quadratic MPC for Converters with a Modulator 563
4 Linear Quadratic Finite Control Set MPC 566
4.1 Closed-Form Solution 567
4.2 Design for Stability and Performance 569
4.3 Example: Reference Tracking 571
5 An Efficient Algorithm for Finite-Control Set MPC 575
5.1 Modified Sphere Decoding Algorithm 576
5.2 Simulation Study of FCS-MPC 579
6 Conclusions 582
References 583
Learning-Based Fast Nonlinear Model Predictive Control for Custom-Made 3D Printed Ground and Aerial Robots 586
1 Introduction 586
2 Receding Horizon Control and Estimation Methods 588
2.1 Nonlinear Model Predictive Control 588
2.2 Nonlinear Moving Horizon Estimation 589
3 Real-Time Applications 591
3.1 Ultra-Compact Field Robot 591
3.1.1 System Description 591
3.1.2 System Model 592
3.1.3 Control Scheme 593
3.1.4 Implementation of NMHE 593
3.1.5 Implementation of NMPC 594
3.1.6 Results 595
3.2 Tilt-Rotor Tricopter UAV 597
3.2.1 System Description 598
3.2.2 System Model 599
3.2.3 Kinematic Equations 599
3.2.4 Rigid-Body Equations 599
3.2.5 External Forces and Moments 600
3.2.6 Control Scheme 602
3.2.7 Implementation of NMPC 602
3.2.8 Implementation of NMHE 603
3.2.9 Results 604
3.2.10 Circular Reference Tracking 604
4 Conclusion 608
References 609
Applications of MPC to Building HVAC Systems 611
1 Introduction to Building HVAC Systems 611
2 Problem Statement 613
2.1 MPC 614
3 Challenges and Opportunities 615
3.1 Modeling 615
3.2 Load Forecasting 616
3.3 Discrete Decisions 617
3.4 Large-Scale Applications 617
3.5 Demand Charges 618
4 Decomposition 618
4.1 High-Level 618
4.2 Low-Level Airside 619
4.3 Low-Level Waterside 619
4.4 Feedback 620
5 Example 620
6 Stanford University Campus 622
6.1 SESI Project 622
6.2 Control System 623
6.3 Performance 624
7 Outlook 624
References 626
Toward Multi-Layered MPC for Complex Electric Energy Systems 628
1 Introduction 628
2 Temporal and Spatial Complexities in the Changing Electric Power Industry 629
3 Load Characterization: The Main Cause of Inter-Temporal Dependencies and Spatial Interdependencies 631
3.1 Multi-Temporal Load Decomposition 634
3.2 Inflexible Load Modeling 634
4 Hierarchical Control in Today's Electric Power Systems 636
4.1 Main Objectives of Hierarchical Control 636
4.2 General Formulation of Main Objectives 638
4.3 Unified Modeling Framework 639
4.4 Assumptions and Limitations Rooted in Today's Hierarchical Control 640
4.4.1 Tertiary Level Control 640
4.4.2 Secondary Level Control 641
4.4.3 Primary Level Control 641
5 Need for Interactive Multi-Layered MPC in Changing Industry 641
5.1 Temporal Aspect 642
5.2 Spatial Aspect 642
6 Temporal Lifting for Decision Making with Multi-Rate Disturbances 643
6.1 Nested Temporal Lifting 644
7 Spatial Lifting for Multi-Agent Decision Making 647
7.1 Nested Spatial Lifting 648
7.1.1 Functional Bids 650
8 Digital Implementation 651
9 Framework for Implementing Interactive Multi-Spatial Multi-Temporal MPC: DyMonDS 653
10 Application of the DyMonDS Framework: One Dayin a Lifetime of Two Bus Power System 655
10.1 Example 1: MPC for Utilizing Heterogeneous Generation Resources 655
10.2 Example 2: MPC Spatial and Temporal Lifting in Microgrids to Support Efficient Participation of Flexible Demand 656
10.3 Example 3: The Role of MPC in Reducing the Need for Fast Storage While Enabling Stable FeedbackResponse 658
10.4 Example 4: The Role of MPC Spatial Lifting in Normal Operation Automatic Generation Control (AGC) 661
11 Conclusions 663
References 663
Applications of MPC to Finance 667
1 Introduction 667
1.1 Portfolio Optimization 667
1.2 Dynamic Option Hedging 668
1.3 Organization of Chapter 669
2 Modeling of Account Value Dynamics 670
2.1 Stock Price Dynamics 672
2.2 Control Structure of Trading Algorithms 673
3 Portfolio Optimization Problems 673
3.1 MPC Formulations 675
3.1.1 Overview of MPC Literature 675
3.1.2 Transaction Costs 677
3.1.3 Constraints in Portfolio Optimization 678
4 MPC in Dynamic Option Hedging 679
4.1 European Call Option Hedging 680
4.2 Option Replication as a Control Problem 681
4.3 MPC Option Hedging Formulations 682
4.3.1 Using an Option Pricing Model 683
4.3.2 Predicting to Expiration 683
4.4 Additional Considerations in Option Hedging 684
5 Conclusions 685
References 685
Index 688