没有合适的资源?快使用搜索试试~ 我知道了~
首页模型预测控制Model Predictive Control: Theory and Design
模型预测控制Model Predictive Control: Theory and Design
需积分: 49 215 下载量 145 浏览量
更新于2023-03-16
评论 6
收藏 3.03MB PDF 举报
MPC,模型预测控制,J.B. Rawlings 和D.Q.Mayne所写。这两位所站的高度和视野广度才能够写出这本优秀的教材来。我日常以这本教材作为辅助参考书使用。
资源详情
资源评论
资源推荐
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
welcome hearing about other papers that the reader feels should be included here.
1
Chapter 1. Getting Started with Model Predictive Control
Offset-free control. In Section 1.5.2, Disturbances and Zero Offset, conditions
are given that ensure zero offset in chosen control variables in the presence of
plant/model mismatch under any choices of stabilizing regulator and stable esti-
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
< p. As
pointed out by Pannocchia and Rawlings (2003), however, choosing n
d
< p also
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
regulator tuning is changed. Maeder et al. (2009) give design procedures for choos-
ing estimator and regulator parameters simultaneously to achieve zero offset in
this situation.
Chapter 2. Model Predictive Control — Regulation
MPC stability results with the KL definition of asymptotic stability. Since Lya-
punov’s foundational work, asymptotic stability traditionally has been defined with
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
1
rawlings@engr.wisc.edu, d.mayne@imperial.ac.uk
1
define state estimator stability in Chapter 4 (see Definition 4.6, for example). We
outline here how to extend the main MPC stability results of Chapter 2 to apply
under this stronger definition of asymptotic stability.
2
In many MPC applications using nonlinear models, it is straightforward to obtain
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
to all
of X
N
, including the case when X
N
is unbounded. Given the local upper bounding
K
∞
function on X
f
, the necessary and sufficient condition for function V(·) to have
an upper bounding K
∞
function on all of X
N
is that V (·) is locally bounded on X
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
0
N
(·) is locally
bounded on X
N
.
Proposition 1 (MPC value function is locally bounded). Suppose Assumptions 2.2
and 2.3 hold. Then V
0
N
(·) is locally bounded on X
N
.
Proof. Let X be an arbitrary compact subset of X
N
. The function V
N
: R
n
× R
Nm
→
R
≥0
is defined and continuous and therefore has an upper bound on the compact
set X × U
N
. Since U
N
(x) ⊂ U
N
for all x ∈ X
N
, V
0
N
: X
N
→ R
≥0
has the same upper
bound on X. Since X is arbitrary, we have established that V
0
N
(·) is locally bounded
on X
N
.
We next extend Proposition 2.18 by removing the assumption that X
N
is com-
pact.
Proposition 2 (Extension of upper bound to X
N
). Suppose that Assumptions 2.2, 2.3,
2.12, and 2.13 hold and that X
f
contains the origin in its interior. If there exists a
K
∞
function α(·) such that V
0
N
(x) ≤ α(|x|) for all x ∈ X
f
, then there exists another
K
∞
function β(·) such that V
0
N
(x) ≤ β(|x|) for all x ∈ X
N
.
Proof. From the definition of X
N
and Assumptions 2.12 and 2.13, we have that
X
f
⊆ X
N
. From Proposition 2.11, we have that the set X
N
is closed, and this
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
also removes the
need to assume X
N
is bounded in Proposition 2.19.
Finally, we can establish Theorem 2.22 under the stronger “KL” definition of
asymptotic stability.
Theorem 4 (Asymptotic stability with unbounded region of attraction). Suppose
X
N
⊂ R
n
and X
f
⊂ X
N
are positive invariant for the system x
+
= f (x), that
X
f
⊂ X
N
is closed and contains the origin in its interior, and that there exist a
function V : R
n
→ R
≥0
and two K
∞
functions α
1
(·) and α
2
(·) such that
V(x) ≥ α
1
(|x|) ∀x ∈ X
N
(1)
V(x) ≤ α
2
(|x|) ∀x ∈ X
f
(2)
V(f (x)) − V (x) ≤ −α
1
(|x|) ∀x ∈ X
N
(3)
2
The authors would like to thank Andy Teel of UCSB for helpful discussion of these issues.
2
Text Postface Summary of change
Proposition 2.18 Proposition 2 Removes boundedness of X
N
Proposition 2.19 Remark 3 Removes boundedness of X
N
Theorem 2.22 Theorem 4 Asymptotic stability with stronger KL definition
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
X
N
for the system x
+
= f (x).
Proof. Proposition 2 extends the local upper bound in (2) to all of X
N
and Theo-
rem 14 then gives asymptotic stability under Definition 13. Both Theorem 14 and
Definition 13 appear in Appendix B of this note.
A summary of these extensions to the results of Chapter 2 and Appendix B is
provided in Table 1.
Positive invariance under control law κ
N
(·). Proposition 2.11 correctly states
that the set X
N
is positive invariant for the closed-loop system x
+
= f (x, κ
N
(x)).
The proof follows from (2.11), and is stated in the text as:
That X
N
is positive invariant for x
+
= f (x, κ
N
(x)) follows from (2.11),
which shows that κ
N
(·) steers every x ∈ X
N
into X
N−1
⊆ X
N
.
But notice that this same argument establishes that X
N−1
is also positive invariant
for the closed-loop system, a fact that does not seem to have been noticed previ-
ously. Since X
N−1
⊆ X
N
, this statement is a tighter characterization of the positive
invariance property. This tighter characterization is sometimes useful when estab-
lishing robust stability for systems with discontinuous V
0
N
(·), such as Example 2.8.
Among the feasibility sets, X
j
, j = 0, 1, . . . , N, the set X
N
is the largest positive
invariant set and X
N−1
is the smallest positive invariant set for x
+
= f (x, κ
N
(x));
none of the other feasibility sets, X
j
, j = 0, 1, . . . , N − 2, are necessarily positive
invariant for x
+
= f (x, κ
N
(x)) for all systems satisfying the given assumptions. A
modified Proposition 2.11 reads as follows.
Proposition 2.11’ (Existence of solutions to DP recursion). Suppose Assumptions
2.2 and 2.3 hold. Then
(a) For all j ∈ I
≥0
, the cost function V
j
(·) is continuous in Z
j
, and, for each x ∈ X
j
,
the control constraint set U
j
(x) is compact and a solution u
0
(x) ∈ U
j
(x) to P
j
(x)
exists.
(b) If X
0
:= X
f
is control invariant for x
+
= f (x, u), u ∈ U, then, for each j ∈ I
≥0
,
the set X
j
is also control invariant, X
j
⊇ X
j−1
, 0 ∈ X
j
, and X
j
is closed.
(c) In addition, the sets X
j
and X
j−1
are positive invariant for x
+
= f (x, κ
j
(x)) for
all j ∈ I
≥1
.
3
Unreachable setpoints, strong duality, and dissipativity. Unreachable setpoints
are discussed in Section 2.9.3. It is known that the optimal MPC value function in
this case is not decreasing and is therefore not a Lyapunov function for the closed-
loop system. A recent paper by Diehl, Amrit, and Rawlings (2011) has shown that
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
stability of economic MPC with nonlinear models. This paper also shows that MPC
is better than optimal periodic control for systems that are not optimally operated
at steady state.
Unbounded input constraint sets. Assumption 2.3 includes the restriction that
the input constraint set U is compact (bounded and closed). This basic assumption
is used to ensure existence of the solution to the optimal control problem through-
out Chapter 2. If one is interested in an MPC theory that handles an unbounded
input constraint set U, then one can proceed as follows. First modify Assumption
2.3 by removing the boundedness assumption on U.
Assumption 5 (Properties of constraint sets – unbounded case). The sets X, X
f
,
and U are closed, X
f
⊆ X; each set contains the origin.
Then, to ensure existence of the solution to the optimal control problem, con-
sider the cost assumption on page 154 in the section on nonpositive definite stage
costs, slightly restated here.
Assumption 6 (Stage cost lower bound). Consider the following two lower bounds
for the stage cost.
(a)
`(y, u) ≥ α
1
(
(y, u)
) for all y ∈ R
p
, u ∈ R
m
V
f
(x) ≤ α
2
(|x|) for all x ∈ X
f
in which α
1
(·) is a K
∞
function.
(b)
`(y, u) ≥ c
1
(y, u)
a
for all y ∈ R
p
, u ∈ R
m
V
f
(x) ≤ c
2
|x|
a
for all x ∈ X
f
in which c
1
, c
2
, a > 0.
Finally, assume that the system is input/output-to-state stable (IOSS). This prop-
erty is given in Definition 2.40 (or Definition B.42). We can then state an MPC sta-
bility theorem that applies to the case of unbounded constraint sets.
Theorem 7 (MPC stability – unbounded constraint sets).
(a) Suppose that Assumptions 2.2, 5, 2.12, 2.13, and 6(a) hold and that the system
x
+
= f (x, u), y = h(x) is IOSS. Then the origin is asymptotically stable (under
Definition 9) with a region of attraction X
N
for the system x
+
= f (x, κ
N
(x)).
4
(b) Suppose that Assumptions 2.2, 5, 2.12, 2.13, and 6(b) hold and that the system
x
+
= f (x, u), y = h(x) is IOSS. Then the origin is exponentially stable with a region
of attraction X
N
for the system x
+
= f (x, κ
N
(x)).
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
Consider the linear, time invariant model x
+
= Ax + Bu, y = Cx with quadratic
penalties `(y, u) = (1/2)(y
0
Qy + u
0
Ru) for both the finite and infinite horizon
cases. What do the assumptions of Theorem 7(b) imply in this case? Compare these
assumptions to the standard LQR assumptions listed in Exercise 1.20 (b).
Assumption 2.2 is satisfied for f (x, u) = Ax + Bu for all A ∈ R
n×n
, B ∈ R
n×m
;
we have X = R
n
, and U = R
m
. 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
f
to be the stabilizable subspace of (A, B) and Assumption 2.13 is satisfied.
The set X
N
contains the system controllability information. The set X
N
is the
stabilizable subspace of (A, B), and we can satisfy Assumption 6(a) by choosing
V
f
(x) = (1/2)x
0
Πx in which Π is the solution to the steady-state Riccati equation
for the stabilizable modes of (A, B).
In particular, if (A, B) is stabilizable, then X
f
= R
n
, X
N
= R
n
for all N ∈ I
0:∞
,
and V
f
can be chosen to be V
f
(x) = (1/2)x
0
Πx in which Π is the solution to the
steady-state Riccati equation (1.19). The horizon N can be finite or infinite with
this choice of V
f
(·) and the control law is invariant with respect to the horizon
length, κ
N
(x) = Kx in which K is the steady-state linear quadratic regulator gain
given in (1.19). Theorem 7(b) then gives that the origin of the closed-loop system
x
+
= f (x, κ
N
(x)) = (A + BK)x is globally, exponentially stable.
The standard assumptions for the LQR with stage cost l(y, u) = (1/2)(y
0
Qy +
u
0
Ru) are
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-
ings, and Wright, 2011) takes the suboptimal MPC formulation in Section 6.1.2,
also discussed in Section 2.8, and establishes its inherent robustness to bounded
process and measurement disturbances. See also the paper by Lazar and Heemels
(2009), which first addressed inherent robustness of suboptimal MPC to process
disturbances by (i) specifying a degree of suboptimality and (ii) using the time-
varying state constraint tightening approach of Lim
´
on Marruedo,
´
Alamo, and Ca-
macho (2002) to achieve recursive feasibility under disturbances.
The key assumption in (Pannocchia et al., 2011) is the following.
5
剩余725页未读,继续阅读
ReginaldZhang
- 粉丝: 7
- 资源: 1
上传资源 快速赚钱
- 我的内容管理 收起
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
会员权益专享
最新资源
- 2022年中国足球球迷营销价值报告.pdf
- 房地产培训 -营销总每天在干嘛.pptx
- 黄色简约实用介绍_汇报PPT模板.pptx
- 嵌入式系统原理及应用:第三章 ARM编程简介_3.pdf
- 多媒体应用系统.pptx
- 黄灰配色简约设计精美大气商务汇报PPT模板.pptx
- 用matlab绘制差分方程Z变换-反变换-zplane-residuez-tf2zp-zp2tf-tf2sos-sos2tf-幅相频谱等等.docx
- 网络营销策略-网络营销团队的建立.docx
- 电子商务示范企业申请报告.doc
- 淡雅灰低面风背景完整框架创业商业计划书PPT模板.pptx
- 计算模型与算法技术:10-Iterative Improvement.ppt
- 计算模型与算法技术:9-Greedy Technique.ppt
- 计算模型与算法技术:6-Transform-and-Conquer.ppt
- 云服务安全风险分析研究.pdf
- 软件工程笔记(完整版).doc
- 电子商务网项目实例规划书.doc
资源上传下载、课程学习等过程中有任何疑问或建议,欢迎提出宝贵意见哦~我们会及时处理!
点击此处反馈
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功
评论0