كتاب Model Predictive Control System Design and Implementation Using MATLAB
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 كتاب Model Predictive Control System Design and Implementation Using MATLAB

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مُساهمةموضوع: كتاب Model Predictive Control System Design and Implementation Using MATLAB    كتاب Model Predictive Control System Design and Implementation Using MATLAB  Emptyالإثنين 29 أغسطس 2022, 11:58 pm

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Model Predictive Control System Design and Implementation Using MATLAB
Liuping Wang  

كتاب Model Predictive Control System Design and Implementation Using MATLAB  M_p_c_11
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Contents
List of Symbols and Abbreviations . xxvii
1 Discrete-time MPC for Beginners . 1
1.1 Introduction 1
1.1.1 Day-to-day Application Example of Predictive Control . 1
1.1.2 Models Used in the Design . 3
1.2 State-space Models with Embedded Integrator 4
1.2.1 Single-input and Single-output System . 4
1.2.2 MATLAB Tutorial: Augmented Design Model . 6
1.3 Predictive Control within One Optimization Window 7
1.3.1 Prediction of State and Output Variables . 7
1.3.2 Optimization . 9
1.3.3 MATLAB Tutorial: Computation of MPC Gains 13
1.4 Receding Horizon Control 15
1.4.1 Closed-loop Control System 16
1.4.2 MATLAB Tutorial: Implementation
of Receding Horizon Control 20
1.5 Predictive Control of MIMO Systems 22
1.5.1 General Formulation of the Model . 22
1.5.2 Solution of Predictive Control for MIMO Systems . 26
1.6 State Estimation 27
1.6.1 Basic Ideas About an Observer 28
1.6.2 Basic Results About Observability . 30
1.6.3 Kalman Filter 33
1.6.4 Tuning Observer Dynamics . 34
1.7 State Estimate Predictive Control . 34
1.8 Summary . 37
Problems . 39xxii Contents
2 Discrete-time MPC with Constraints 43
2.1 Introduction 43
2.2 Motivational Examples . 43
2.3 Formulation of Constrained Control Problems . 47
2.3.1 Frequently Used Operational Constraints . 47
2.3.2 Constraints as Part of the Optimal Solution 50
2.4 Numerical Solutions Using Quadratic Programming 53
2.4.1 Quadratic Programming for Equality Constraints 53
2.4.2 Minimization with Inequality Constraints . 58
2.4.3 Primal-Dual Method . 62
2.4.4 Hildreth’s Quadratic Programming Procedure . 63
2.4.5 MATLAB Tutorial: Hildreth’s Quadratic Programming 67
2.4.6 Closed-form Solution of λ∗ . 68
2.5 Predictive Control with Constraints on Input Variables . 69
2.5.1 Constraints on Rate of Change 70
2.5.2 Constraints on Amplitude of the Control . 73
2.5.3 Constraints on Amplitude and Rate of Change 77
2.5.4 Constraints on the Output Variable 78
2.6 Summary . 81
Problems . 83
3 Discrete-time MPC Using Laguerre Functions . 85
3.1 Introduction 85
3.2 Laguerre Functions and DMPC . 85
3.2.1 Discrete-time Laguerre Networks 86
3.2.2 Use of Laguerre Networks in System Description 90
3.2.3 MATLAB Tutorial: Use of Laguerre Functions
in System Modelling . 90
3.3 Use of Laguerre Functions in DMPC Design 92
3.3.1 Design Framework . 93
3.3.2 Cost Functions 94
3.3.3 Minimization of the Objective Function 97
3.3.4 Convolution Sum 98
3.3.5 Receding Horizon Control 98
3.3.6 The Optimal Trajectory of Incremental Control . 99
3.4 Extension to MIMO Systems 106
3.5 MATLAB Tutorial Notes . 108
3.5.1 DMPC Computation . 108
3.5.2 Predictive Control System Simulation 115
3.6 Constrained Control Using Laguerre Functions 118
3.6.1 Constraints on the Difference of the Control Variable 118
3.6.2 Constraints on the Amplitudes of the Control Signal . 121
3.7 Stability Analysis 127
3.7.1 Stability with Terminal-State Constraints 127
3.7.2 Stability with Large Prediction Horizon 129Contents xxiii
3.8 Closed-form Solution of Constrained Control for SISO Systems 131
3.8.1 MATLAB Tutorial: Constrained Control of DC Motor . 135
3.9 Summary . 143
Problems . 144
4 Discrete-time MPC with Prescribed Degree of Stability . 149
4.1 Introduction 149
4.2 Finite Prediction Horizon: Re-visited 149
4.2.1 Motivational Example . 150
4.2.2 Origin of the Numerical Conditioning Problem 150
4.3 Use of Exponential Data Weighting 152
4.3.1 The Cost Function 152
4.3.2 Optimization of Exponentially Weighted Cost Function 153
4.3.3 Interpretation of Results from Exponential Weighting 156
4.4 Asymptotic Closed-loop Stability with Exponential Weighting . 158
4.4.1 Modification of Q and R Matrices . 158
4.4.2 Interpretation of the Results 160
4.5 Discrete-time MPC with Prescribed Degree of Stability . 165
4.6 Tuning Parameters for Closed-loop Performance . 170
4.6.1 The Relationship Between P∞ and Jmin 171
4.6.2 Tuning Procedure Once More . 176
4.7 Exponentially Weighted Constrained Control 179
4.8 Additional Benefit . 182
4.9 Summary . 186
Problems . 188
5 Continuous-time Orthonormal Basis Functions . 193
5.1 Introduction 193
5.2 Orthonormal Expansion 193
5.3 Laguerre Functions 194
5.4 Approximating Impulse Responses . 197
5.5 Kautz Functions . 202
5.5.1 Kautz Functions in the Time Domain 204
5.5.2 Modelling the System Impulse Response 205
5.6 Summary . 206
Problems . 207
6 Continuous-time MPC . 209
6.1 Introduction 209
6.2 Model Structures for CMPC Design 209
6.2.1 Model Structure . 211
6.2.2 Controllability and Observability of the Model 215
6.3 Model Predictive Control Using Finite Prediction Horizon 216
6.3.1 Modelling the Control Trajectory 217
6.3.2 Predicted Plant Response 218xxiv Contents
6.3.3 Analytical Solution of the Predicted Response 219
6.3.4 The Recursive Solution . 221
6.4 Optimal Control Strategy 224
6.5 Receding Horizon Control 227
6.6 Implementation of the Control Law in Digital Environment . 234
6.6.1 Estimation of the States 234
6.6.2 MATLAB Tutorial: Closed-loop Simulation . 237
6.7 Model Predictive Control Using Kautz Functions 240
6.8 Summary . 244
Problems . 245
7 Continuous-time MPC with Constraints . 249
7.1 Introduction 249
7.2 Formulation of the Constraints 249
7.2.1 Frequently Used Constraints 249
7.2.2 Constraints as Part of the Optimal Solution 251
7.3 Numerical Solutions for the Constrained Control Problem 257
7.4 Real-time Implementation of Continuous-time MPC . 262
7.5 Summary . 266
Problems . 267
8 Continuous-time MPC with Prescribed Degree of Stability 271
8.1 Introduction 271
8.2 Motivating Example . 271
8.3 CMPC Design Using Exponential Data Weighting . 274
8.4 CMPC with Asymptotic Stability . 277
8.5 Continuous-time MPC with Prescribed Degree of Stability 283
8.5.1 The Original Anderson and Moore’s Results 283
8.5.2 CMPC with a Prescribed Degree of Stability 284
8.5.3 Tuning Parameters and Design Procedure 286
8.6 Constrained Control with Exponential Data Weighting . 288
8.7 Summary . 291
Problems . 293
9 Classical MPC Systems in State-space Formulation 297
9.1 Introduction 297
9.2 Generalized Predictive Control in State-space Formulation 298
9.2.1 Special Class of Discrete-time State-space Structures . 298
9.2.2 General NMSS Structure for GPC Design 301
9.2.3 Generalized Predictive Control in State-space
Formulation 302
9.3 Alternative Formulation to GPC 305
9.3.1 Alternative Formulation for SISO Systems 305
9.3.2 Closed-loop Poles of the Predictive Control System 307
9.3.3 Transfer Function Interpretation 310Contents xxv
9.4 Extension to MIMO Systems 313
9.4.1 MNSS Model for MIMO Systems 314
9.4.2 Case Study of NMSS Predictive Control System . 315
9.5 Continuous-time NMSS model 320
9.6 Case Studies for Continuous-time MPC 323
9.7 Predictive Control Using Impulse Response Models 326
9.8 Summary . 329
Problems . 330
10 Implementation of Predictive Control Systems . 333
10.1 Introduction 333
10.2 Predictive Control of DC Motor Using a Micro-controller . 333
10.2.1 Hardware Configuration 334
10.2.2 Model Development . 336
10.2.3 DMPC Tuning 337
10.2.4 DMPC Implementation 338
10.2.5 Experimental Results 339
10.3 Implementation of Predictive Control Using xPC Target 340
10.3.1 Overview . 340
10.3.2 Creating a SIMULINK Embedded Function . 342
10.3.3 Constrained Control of DC Motor Using xPC Target . 347
10.4 Control of Magnetic Bearing Systems 349
10.4.1 System Identification 351
10.4.2 Experimental Results 352
10.5 Continuous-time Predictive Control of Food Extruder 353
10.5.1 Experimental Setup . 355
10.5.2 Mathematical Models 357
10.5.3 Operation of the Model Predictive Controller . 358
10.5.4 Controller Tuning Parameters . 359
10.5.5 On-line Control Experiments 360
10.6 Summary . 365
References . 367
Index
List of Symbols and Abbreviations
Symbols
a Scaling factor for discrete-time Laguerre functions
arg min Minimizing argument
A State matrix of state-space model
B Input-to-state matrix of state-space model
C State-to-output matrix of state-space model
D Direct feed-through matrix of state-space model
(A, B, C, D) State-space realization
ΔU Parameter vector for the control sequence
Δu(ki + m) future incremental control at sample m
Δumin, Δumax Minimum and maximum limits for Δu
F, Φ Pair of matrices used in the prediction equation Y =
Fx(ki) + ΦΔU
G(s) Transfer function model
Iq
×q Identity matrix with appropriate dimensions
J Performance index for optimization
Klqr Feedback control gain using LQR
K
mpc Feedback control gain using MPC
Kx State feedback control gain vector related to Δxm(·) or
x˙ m(·)
Ky
State feedback control gain related to y
Kob Observer gain vector
κ(A) Condition number of A matrix
li(·) The ith discrete or continuous-time Laguerre function
L(·) Discrete and continuous-time Laguerre functions in vector
form
Li(s) Laplace transform of the ith continuous-time Laguerre
function
Li(z) z-transform of the ith discrete Laguerre function
λ Lagrange multiplierxxviii List of Symbols and Abbreviations
λi(A) The ith eigenvalue of matrix A
m Number of inputs, also the mth future sample in discrete
time
M, γ Pair of matrix, vector for inequality constraints (Mx ≤ γ)
N Number of terms used in Laguerre function expansion, both
continuous and discrete time
Nc Control horizon
Np
Prediction horizon
om Zero vector with appropriate dimension
ok Zero row vector (k = 1, 2, .) with appropriate dimensions
o
q×q q × q zero matrix
o
q×m q × m zero matrix
Ω, Ψ Pair of matrices in the cost of predictive control J =
ηTΩη + 2ηTΨx(·) + cons
η Parameter vector in the Laguerre expansion
p Scaling factor for continuous-time Laguerre functions
Q, R Pair of weight matrices in the cost function of predictive
control
q−i Backward shift operator, q−i[f(k)] = f(k − i)
q Number of outputs
r(·) Set-point signal
Sact Index set of active constraints
Tp
Prediction horizon in continuous-time
u(·) Control signal
umin, umax Minimum and maximum limits for u
x(·) State variable
x(ki + m | ki) Predicted state variable vector at sample time m, given
current state x(ki)
x(ti + τ | ti) Predicted state variable vector at time τ given current state
x(ti)
xˆ(·) Estimated state variable vector in both continuous and
discrete-time
y(·) Output signal
Y Predicted output data vector
ymin, ymax Minimum and maximum limits for y
Abbreviations
CMPC Continuous-time model predictive control
DLQR Discrete-time linear quadratic regulatorList of Symbols and Abbreviations xxix
DMPC Discrete-time model predictive control
FIR Finite impulse response
LQR Linear quadratic regulator
MIMO Multiple-input, multiple-output
SISO Single-input, single-output
List of Symbols and Abbreviations
Symbols
a Scaling factor for discrete-time Laguerre functions
arg min Minimizing argument
A State matrix of state-space model
B Input-to-state matrix of state-space model
C State-to-output matrix of state-space model
D Direct feed-through matrix of state-space model
(A, B, C, D) State-space realization
ΔU Parameter vector for the control sequence
Δu(ki + m) future incremental control at sample m
Δumin, Δumax Minimum and maximum limits for Δu
F, Φ Pair of matrices used in the prediction equation Y =
Fx(ki) + ΦΔU
G(s) Transfer function model
Iq
×q Identity matrix with appropriate dimensions
J Performance index for optimization
Klqr Feedback control gain using LQR
K
mpc Feedback control gain using MPC
Kx State feedback control gain vector related to Δxm(·) or
x˙ m(·)
Ky
State feedback control gain related to y
Kob Observer gain vector
κ(A) Condition number of A matrix
li(·) The ith discrete or continuous-time Laguerre function
L(·) Discrete and continuous-time Laguerre functions in vector
form
Li(s) Laplace transform of the ith continuous-time Laguerre
function
Li(z) z-transform of the ith discrete Laguerre function
λ Lagrange multiplierxxviii List of Symbols and Abbreviations
λi(A) The ith eigenvalue of matrix A
m Number of inputs, also the mth future sample in discrete
time
M, γ Pair of matrix, vector for inequality constraints (Mx ≤ γ)
N Number of terms used in Laguerre function expansion, both
continuous and discrete time
Nc Control horizon
Np
Prediction horizon
om Zero vector with appropriate dimension
ok Zero row vector (k = 1, 2, .) with appropriate dimensions
o
q×q q × q zero matrix
o
q×m q × m zero matrix
Ω, Ψ Pair of matrices in the cost of predictive control J =
ηTΩη + 2ηTΨx(·) + cons
η Parameter vector in the Laguerre expansion
p Scaling factor for continuous-time Laguerre functions
Q, R Pair of weight matrices in the cost function of predictive
control
q−i Backward shift operator, q−i[f(k)] = f(k − i)
q Number of outputs
r(·) Set-point signal
Sact Index set of active constraints
Tp
Prediction horizon in continuous-time
u(·) Control signal
umin, umax Minimum and maximum limits for u
x(·) State variable
x(ki + m | ki) Predicted state variable vector at sample time m, given
current state x(ki)
x(ti + τ | ti) Predicted state variable vector at time τ given current state
x(ti)
xˆ(·) Estimated state variable vector in both continuous and
discrete-time
y(·) Output signal
Y Predicted output data vector
ymin, ymax Minimum and maximum limits for y
Abbreviations
CMPC Continuous-time model predictive control
DLQR Discrete-time linear quadratic regulatorList of Symbols and Abbreviations xxix
DMPC Discrete-time model predictive control
FIR Finite impulse response
LQR Linear quadratic regulator
MIMO Multiple-input, multiple-output
SISO Single-input, single-output

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