英文版-模型预测控制理论与设计.pdf

Postface to “Model Predictive Control: Theory

and Design”

J. B. Rawlings and D. Q. Mayne

August 19, 2012

The goal of this postface is to point out and comment upon recent MPC papers

and issues pertaining to topics covered in the first printing of the monograph by

Rawlings and Mayne (2009). We have tried to group the recent MPC literature by the

relevant chapter in that reference. This compilation is selective and not intended

to be a comprehensive summary of the current MPC research literature, but we

mator. In particular, choosing the number of integrating disturbances equal to the

number of measurements, n

d

= p, achieves zero offset independently of estimator

and regulator tuning. A recent contribution by Maeder, Borrelli, and Morari (2009)

tackles the issue of achieving offset free performance when choosing n

d

pointed out by Pannocchia and Rawlings (2003), however, choosing n

means that the gain of the estimator depends on the regulator tuning. Therefore,

to maintain offset free performance, the estimator tuning must be changed if the

two fundamental conditions: (i) local stability and (ii) attractivity. Control and sys-

tems texts using this classical definition include Khalil (2002, p. 112) and Vidyasagar

(1993, p. 141). The classical definition was used mainly in stating and proving the

stability theorems appearing in the Appendix B corresponding to the first printing

of the text. Recently, however, a stronger definition of asymptotic stability, which

we refer to here as the “KL” definition, has started to become popular. These two

definitions are compared and contrasted in a later section of this postface (see Ap-

pendix B – Stability Theory). We used the KL definition of asymptotic stability to

an upper bound on the MPC value function on a small set X

f

containing the origin

in its interior. For example, this bound can be established when the linearization of

the system is stabilizable at the origin. But it may be difficult to extend this upper

bound to cover the entire stabilizable set X

N

. But we require this upper bound to

apply standard Lyapunov stability theory to the MPC controller. Therefore, we next

wish to extend the upper bounding K

function α

2

(·) from the local set X

f

N

N

i.e., V (·) is bounded on every compact subset of X

N

. See Appendix B of this note

for a statement and proof of this result. So we first establish that V

bounded on X

Proposition 1 (MPC value function is locally bounded). Suppose Assumptions 2.2

and 2.3 hold. Then V

(·) is locally bounded on X

.

≥0

set X × U

N

. Since U

(x) ⊂ U

for all x ∈ X

: X

N

≥0

has the same upper

bound on X. Since X is arbitrary, we have established that V

0

N

(·) is locally bounded

N

2.12, and 2.13 hold and that X

f

function β(·) such that V

N

f

N

N

proposition therefore follows directly from Proposition 11 in Appendix B of this

note. 

Remark 3. The extension of Proposition 2.18 to unbounded X

N

need to assume X

Finally, we can establish Theorem 2.22 under the stronger “KL” definition of

asymptotic stability.

f

function V : R

n

≥0

and two K

functions α

1

V(x) ≥ α

N

V(x) ≤ α

f

V(f (x)) − V (x) ≤ −α

1

N

Definition B.6 Definition 9 Classical to KL definition of asymptotic stability

Theorem B.11 Theorem 12 Lyapunov function and KL definition

Definition B.9 (e) Definition 13 Asymptotic stability with KL definition (constrained)

Theorem B.13 Theorem 14 Lyapunov function and KL definition (constrained)

Table 1: Extensions of MPC stability results in Chapter 2 and Appendix B.

Then the origin is asymptotically stable under Definition 9 with a region of attraction

for the system x

= f (x).

Proof. Proposition 2 extends the local upper bound in (2) to all of X

and Theo-

a modified MPC cost function, termed rotated cost, is a Lyapunov function for the

unreachable setpoint case and other more general cost functions required for op-

timizing process economics. A strong duality condition is shown to be a sufficient

condition for asymptotic stability of economic MPC with nonlinear models.

This result is further generalized in the recent paper Angeli, Amrit, and Rawl-

ings (2012). Here a dissipation inequality is shown to be sufficient for asymptotic

x

= f (x, u), y = h(x) is IOSS. Then the origin is asymptotically stable (under

Definition 9) with a region of attraction X

= f (x, κ

N

In particular, setting up the MPC theory with these assumptions subsumes the

LQR problem as a special case.

Example 1: The case of the linear quadratic regulator

Assumption 2.2 is satisfied for f (x, u) = Ax + Bu for all A ∈ R

;

we have X = R

n

. Assumption 6(b) implies that Q > 0 and R > 0.

The system being IOSS implies that (A, C) is detectable (see Exercise 4.5). We can

choose X

The set X

stabilizable subspace of (A, B), and we can satisfy Assumption 6(a) by choosing

n

N

and V

f

(x) = (1/2)x

steady-state Riccati equation (1.19). The horizon N can be finite or infinite with

this choice of V

f

length, κ

u

Q > 0 R > 0 (A, C) detectable (A, B) stabilizable

and we see that this case is subsumed by Theorem 7(b). 

Chapter 6. Distributed Model Predictive Control

The recent paper (Stewart, Venkat, Rawlings, Wright, and Pannocchia, 2010) pro-

vides a compact treatment of many of the issues and results discussed in Chapter

6. Also, for plants with sparsely coupled input constraints, it provides an extension

that achieves centralized optimality on convergence of the controllers’ iterations.

Suboptimal MPC and inherent robustness. The recent paper (Pannocchia, Rawl-

macho (2002) to achieve recursive feasibility under disturbances.

The key assumption in (Pannocchia et al., 2011) is the following.