"模型预测控制理论与设计：Rawlings与Mayne著"

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《模型预测控制理论与设计》是J.B. Rawlings和D.Q.Mayne合著的教材，涵盖了模型预测控制（MPC）的理论和设计。作者分别来自美国威斯康辛大学麦迪逊分校的化学与生物工程系和英国伦敦帝国理工学院的电子工程系。该书由Nob Hill Publishing出版，使用LATEX排版，使用了Forest Stewardship Council认证的无酸再生纸印刷精装。该书深入探讨了MPC的各个方面，是MPC领域的一本重要参考书籍。

List of Figures

1.1 System with input u, output y and transfer function ma-

trix G connecting them; the model is y = Gu. . . . . . . . . 4

1.2 Output of a stochastic system versus time. . . . . . . . . . . 8

1.3 Two quadratic functions and their sum. . . . . . . . . . . . . 14

1.4 Schematic of the moving horizon estimation problem. . . . 38

1.5 MPC controller consisting of: receding horizon regulator,

state estimator, and target selector. . . . . . . . . . . . . . . 50

1.6 Schematic of the well-stirred reactor. . . . . . . . . . . . . . . 52

1.7 Three measured outputs versus time after a step change

in inlet ﬂowrate at 10 minutes; n

= 2. . . . . . . . . . . . . 55

1.8 Two manipulated inputs versus time after a step change

in inlet ﬂowrate at 10 minutes; n

= 2. . . . . . . . . . . . . 56

1.9 Three measured outputs versus time after a step change

in inlet ﬂowrate at 10 minutes; n

= 3. . . . . . . . . . . . . 57

1.10 Two manipulated inputs versus time after a step change

in inlet ﬂowrate at 10 minutes; n

= 3. . . . . . . . . . . . . 58

1.11 Plug-ﬂow reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1.12 Pendulum with applied torque. . . . . . . . . . . . . . . . . . 62

1.13 Feedback control system with output disturbance d, and

setpoint y

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.1 Example of MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2 Feasible region U

, elliptical cost contours, and ellipse

center, a(x), and constrained minimizers for different

values of x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.3 First element of control constraint set U

(x) (shaded)

and control law κ

(x) (circle) versus x = (cos(θ), sin(θ)),

θ ∈ [−π, π] on the unit circle for a nonlinear system with

terminal constraint. . . . . . . . . . . . . . . . . . . . . . . . . 105

2.4 Optimal cost V

(x) versus x on the unit circle. . . . . . . . 106

2.5 The sets Z

and X

. . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.6 Closed-loop performance of sp-MPC and targ-MPC. . . . . . 166

2.7 Region of attraction (shaded region) for constrained MPC

controller of Exercise 2.6. . . . . . . . . . . . . . . . . . . . . . 174

xxiii

xxiv List of Figures

2.8 The region X

, in which the unconstrained LQR control

law is feasible for Exercise 2.7. . . . . . . . . . . . . . . . . . . 175

2.9 The region of attraction for terminal constraint x(N) ∈

and terminal penalty V

(x) = (1/2)x

Πx and the es-

timate of

for Exercise 2.8. . . . . . . . . . . . . . . . . . . 176

2.10 Inconsistent setpoint (x

, u

), unreachable stage cost

`(x, u), and optimal steady states (x

, u

), and stage costs

(x, u) for constrained and unconstrained systems. . . . 180

2.11 Stage cost versus time for the case of unreachable setpoint. 181

3.1 Open-loop and feedback trajectories. . . . . . . . . . . . . . . 192

3.2 The sets X

, R

, and R

. . . . . . . . . . . . . . . . . . . . . . 207

3.3 State trajectory and state tube. . . . . . . . . . . . . . . . . . 219

3.4 Outer-bounding tube X(z, v). . . . . . . . . . . . . . . . . . . . 226

3.5 Tube for a nonlinear system. . . . . . . . . . . . . . . . . . . . 250

3.6 Comparison of 100 realizations of standard and tube-

based MPC for the chemical reactor example. . . . . . . . . 252

3.7 Comparison of standard and tube-based MPC with an ag-

gressive ancillary controller. . . . . . . . . . . . . . . . . . . . 253

3.8 Concentration versus time for the ancillary controller for

sample time 12 (top) and 8 (bottom). . . . . . . . . . . . . . . 254

3.9 Closed-loop robust MPC state evolution with |w| ≤ 0.1

from four different x

. . . . . . . . . . . . . . . . . . . . . . . 259

4.1 MHE arrival cost

(p), underbounding prior weighting

(p), and MHE optimal value

. . . . . . . . . . . . . . . . . 282

4.2 Concatenating two MHE sequences to create a single state

estimate sequence. . . . . . . . . . . . . . . . . . . . . . . . . . 285

4.3 Smoothing update. . . . . . . . . . . . . . . . . . . . . . . . . . 291

4.4 Evolution of the state (solid line) and EKF state estimate

(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

4.5 Evolution of the state (solid line) and UKF state estimate

(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

4.6 Evolution of the state (solid line) and MHE state estimate

(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

4.7 Exact and sampled density. . . . . . . . . . . . . . . . . . . . . 302

4.8 Sampled and exact probability densities for ξ ∼ N(0, 1);

5000 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

4.9 Sampled and exact probability densities for nonlinear trans-

formation η = e

; 5000 samples. . . . . . . . . . . . . . . . . 306

List of Figures xxv

4.10 Probability density Pr(P

(x; s)) for x corresponding to

P(x) = 0.5 and s = 5 samples. . . . . . . . . . . . . . . . . . . 308

4.11 Probability density Pr(P

(x; s)) for three different x cor-

responding to P(x) = 0.05, 0.5, 0.95 and s = 50 samples. . 309

4.12 Probability density Pr(P

(x; s)) for s = 25 samples at

three different x. . . . . . . . . . . . . . . . . . . . . . . . . . . 311

4.13 Probability density Pr(P

(x; s)) for s = 250 samples. . . . . 311

4.14 Cumulative distribution for the sampling error Pr(D

) for

three different sample sizes . . . . . . . . . . . . . . . . . . . 312

4.15 Probability density p(x) to be sampled and the corre-

sponding cumulative distribution P(x). . . . . . . . . . . . . 318

4.16 Six samples of the uniform density on [0, 1], u

, and the

corresponding samples of p(x), x

. . . . . . . . . . . . . . . 318

4.17 Importance function q(x) and its histogram based on

5000 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

4.18 Exact density p(x) and its histogram based on 5000 im-

portance samples. . . . . . . . . . . . . . . . . . . . . . . . . . 321

4.19 Exact density p(x) and its histogram based on 5000 im-

portance samples evaluating h(x) in place of p(x) =

h(x)/

h(x)dx. . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

4.20 Interval [0, 1] partitioned by original sample weights, w

. . 325

4.21 Resampled density of Example 4.33 using 500 samples. . . 329

4.22 Particles’ locations versus time for the simplest particle

ﬁlter; 250 particles. . . . . . . . . . . . . . . . . . . . . . . . . . 336

4.23 Particles’ locations versus time for the simplest particle

ﬁlter with resampling; 250 particles. . . . . . . . . . . . . . . 338

4.24 Fraction of particles inside the 95% contour of the true

conditional density versus time; with and without resam-

pling; average of 500 runs. . . . . . . . . . . . . . . . . . . . . 339

4.25 Particles’ locations versus time using the optimal impor-

tance function; 250 particles. . . . . . . . . . . . . . . . . . . . 346

4.26 Particles’ locations versus time using the optimal impor-

tance function with resampling; 250 particles. . . . . . . . . 347

4.27 Fraction of particles inside the 95% contour of the true

conditional density versus time; with and without resam-

pling; average of 500 runs. . . . . . . . . . . . . . . . . . . . . 348

4.28 Pure MHE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

4.29 Pure PF with optimal importance function. . . . . . . . . . . 353

4.30 Combination MHE/PF with simple importance function. . . 354

4.31 Combination MHE/PF with optimal importance function. . 355

xxvi List of Figures

5.1 State estimator tube. . . . . . . . . . . . . . . . . . . . . . . . . 374

5.2 State tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

5.3 The sets Φ(i), Σ(i) and S(i). . . . . . . . . . . . . . . . . . . . 386

6.1 Convex step from (u

, u

) to (u

p+1

, u

p+1

). . . . . . . . . . 426

6.2 Ten iterations of noncooperative steady-state calculation. . 443

6.3 Ten iterations of cooperative steady-state calculation. . . . 443

6.4 Ten iterations of noncooperative steady-state calculation. . 444

6.5 Ten iterations of cooperative steady-state calculation. . . . 444

6.6 Cooperative control stuck on the boundary of U under

coupled constraints . . . . . . . . . . . . . . . . . . . . . . . . 452

6.7 Cost contours for a two-player, nonconvex game. . . . . . . 462

6.8 Optimizing a quadratic function in one set of variables at

a time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

6.9 Constrained optimality conditions and the normal cone. . 477

7.1 The sets Z, X and U(x). . . . . . . . . . . . . . . . . . . . . . 486

7.2 Parametric linear program. . . . . . . . . . . . . . . . . . . . . 486

7.3 Unconstrained parametric quadratic program. . . . . . . . . 487

7.4 Parametric quadratic program. . . . . . . . . . . . . . . . . . . 487

7.5 Polar cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

7.6 Regions R

, x ∈ X for a second-order example; from

Mayne and Rakovi

c (2003) . . . . . . . . . . . . . . . . . . . . 500

7.7 Regions R

, x ∈ X for a second-order example. . . . . . . . 501

7.8 Solution to a parametric linear program. . . . . . . . . . . . 511

A.1 The four fundamental subspaces of matrix A . . . . . . . . 536

A.2 Matrix A maps into R(A). . . . . . . . . . . . . . . . . . . . . . 537

A.3 Pseudo-inverse of A maps into R(A

). . . . . . . . . . . . . . 537

A.4 Subgradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

A.5 Separating hyperplane. . . . . . . . . . . . . . . . . . . . . . . 552

A.6 Polar cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

A.7 A convex function. . . . . . . . . . . . . . . . . . . . . . . . . . 556

A.8 Normal distribution. . . . . . . . . . . . . . . . . . . . . . . . . 568

A.9 Multivariate normal in two dimensions. . . . . . . . . . . . . 570

A.10 The geometry of quadratic form x

Ax = b. . . . . . . . . . . 571

A.11 A nearly singular normal density in two dimensions. . . . . 575

A.12 The region X(c) for y = max(x

, x

) ≤ c. . . . . . . . . . . . 577

A.13 A joint density function for the two uncorrelated random

variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

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