显式非线性模型预测控制理论与应用详解

模型预测控制

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"《显式非线性模型预测控制理论与应用》是一本深入探讨模型预测控制技术在非线性系统中的最新著作。该书由来自德国汉诺威大学、斯图加特大学和瑞士苏黎世联邦理工学院的三位专家——曼弗雷德·托马博士、弗兰克·奥尔格沃博士和曼弗雷德·莫拉里博士共同编撰。他们分别在控制系统理论、系统理论和自动控制领域拥有深厚的专业背景，这使得本书的内容涵盖了理论研究和实际应用的广度。本书作为《控制与信息科学讲座系列》第429卷的一部分，汇集了国际上对模型预测控制领域的前沿研究成果。系列顾问板包括来自英国谢菲尔德大学、美国加州大学圣巴巴拉分校、俄罗斯莫斯科国立大学、荷兰乌特勒支大学、瑞典隆德理工大学和美国麻省理工学院的多位知名学者，他们的指导和认可为本书的质量提供了保障。书中主要探讨了显式非线性模型预测控制（explicit nonlinear model predictive control, ENMPC）的概念和方法，包括但不限于：非线性动态系统的建模、预测模型的选择和设计、优化算法在求解预测控制问题中的应用、以及实时性和计算效率的权衡。ENMPC特别强调了在处理复杂工业过程和动态环境中，如何通过精确的预测来实现高效控制策略，同时考虑到系统的约束和不确定性。作者们还分享了他们在诸如电力系统、航空航天、机器人技术、化学工程等领域的实践经验，展示了ENMPC在解决实际问题中的有效性。此外，书中还可能包含对未来研究方向的展望，以及对于初学者和研究人员进一步探索非线性模型预测控制技术的指导。总体来说，《显式非线性模型预测控制理论与应用》不仅提供了深入的理论基础，也提供了实用的工具和技术，是理解和掌握现代工业自动化中关键控制技术的重要参考资料。无论是对理论研究者还是工程师，这本书都是提升控制理论和实践能力的一部不可多得的宝典。"

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1.1 Multi-parametric Nonlinear Programming 3

GivenanindexsetA ⊆{1, 2, ... ,q},thecritical region CR

is the set of parame-

ters for which the optimal active set is equal to A , i.e.:

 {x ∈X | A

∗

(x)=A } . (1.6)

As it will be shown in Section 1.2, for strictly convex quadratic function f and

linear constraints g, the critical regionsCR

are polyhedrons and the optimizer z

∗

unique, piecewise afﬁne, and continuous. However, for general nonlinear functions

f and g, the exact solution of the multi-parametric programming problem (1.1)–(1.2)

can not be found, and suboptimal methods for approximating its optimizer function

∗

(x) (or selection in case the optimizer function is not unique) are described in

Section 1.1.5.

1.1.2 Optimality Conditions

For a given x

∈ X, a local minimum z

of problem (1.1)–(1.2) has to satisfy the

well known Karush-Kuhn-Tucker (KKT) ﬁrst-order conditions [56]:

∇

L(z

)=0 (1.7)

diag(

)g(z

)=0 (1.8)

≥ 0 (1.9)

g(z

) ≤ 0 , (1.10)

with associated Lagrange multiplier

and the Lagrangian deﬁned as:

L(z,x,

)  f (z,x)+

g(z,x) . (1.11)

Here, sufﬁcient regularity (smoothness) is assumed, and this will be discussed later

in Section 1.1.4.

Consider the optimal active set A

at x

, i.e. a set of indices to active constraints

in (1.10). The above conditions are sufﬁcient provided the following second order

condition holds [56]:

∇

L(z

)v > 0, ∀v ∈ F −{0} (1.12)

with F being the set of all directions where it is not given from the ﬁrst order

conditions if the objective function will increase or decrease:

F = {v ∈ R

| ∇

)v ≥ 0,

∇

)v = 0, for all i with (

)

> 0} . (1.13)

The notation g

means the rows of g with indices in A

4 1 Multi-parametric Programming

1.1.3 Nonlinear Programming Methods

There exist various methods to numerically compute a local minimum z

of the

problem (1.1)–(1.2) for a given x

∈ X. The most commonly used methods can be

classiﬁed in the following three groups.

1.1.3.1 Newton-Type Methods

The Newton type methods [20] appear to be the most widely used optimization

methods. They try to ﬁnd a point satisfying the KKT conditions (1.7)–(1.10) by

using successive linearizations of the problem functions. The motivation behind

this is that the linearized KKT system can be solved by using standard numeric

linear algebra tools. Depending on how the conditions (1.8)–(1.10) (related to the

imposed constraints) are treated, the two main groups of Newton type methods are

the Sequential Quadratic Programming (SQP) methods and the Interior Point (IP)

methods.

• Sequential Quadratic Programming (SQP) methods.

The SQP methods iteratively solve the KKT system (1.7)–(1.10) by linearizing

the nonlinear functions included in it. The resulting linearized KKT system at

the k + 1-th iteration can be considered as the KKT conditions of the following

quadratic program (QP):

∗

)=min

(z,z

) (1.14)

s.t. g(z

)+∇

g(z

)(z −z

) ≤ 0 , (1.15)

with the quadratic objective function given by:

(z,z

)=∇

f (z

)

z +

(z −z

)

∇

L(z

)(z −z

) . (1.16)

Here, z

and

represent, respectively, the values of optimization variables and

Lagrange multipliers, which solve the k-th sequential iteration of the KKT sys-

tem (1.7)–(1.10). It is assumed that an initial guess z

is provided. In the case

when the Hessian matrix ∇

L(z

) is positive semi-deﬁnite, the QP prob-

lem (1.14)–(1.16) is convex and its unique solution can be found.

Typically, the QP sub-problem (1.14)–(1.16) is solved by using an Active

Set (AS) method [56, 49], which identiﬁes the active set of its solution z

∗

The method begins with ﬁnding a feasible initial guess A

(z,z

)={i ∈

{1,2, ...,q}|g

)+∇

)(z −z

)=0} of the active set by solving

a linear programming problem [56]. In the next iteration, A

(z,z

) is re-

ﬁned by deleting a constraint from A

(z,z

) or by adding a constraint to

(z,z

). In this way, the active set is reﬁned iteratively until the optimal

active set A

∗

) is found.

There are several SQP methods which use approximations of the Hessian

matrix ∇

L(z

) and the constraints Jacobian matrix ∇

g(z

),andthey

1.1 Multi-parametric Nonlinear Programming 5

are referred to as quasi-Newton methods. They usually lead to slower conver-

gence rates, but computationally less expensive iterations, in comparison to the

exact SQP method. One of the quasi-Newton SQP methods is the method by

Powell [61]. It uses exact constraints Jacobian matrix, but replaces the Hessian

matrix ∇

L(z

) by an approximation H

. Each new Hessian approxima-

tion H

k+1

is obtained from the previous approximation H

by an update formula

that uses the difference of the Lagrange gradients,

= ∇

L(z

k+1

) −

∇

L(z

k+1

), and the step

= z

k+1

−z

in order to obtain second order

information in H

k+1

. The most widely used update formula is the one by Broyden-

Fletcher-Goldfarb-Shanno (BFGS) [56]:

k+1

= H

ψψ

−

ττ

. (1.17)

Another successful quasi-Newton SQP method is the constrained Gauss-Newton

method [21]. It uses approximations of the Hessian matrix, based on some Jaco-

bian, and is applicable when the objective function is a sum of squares.

• Interior Point (IP) methods.

The IP methods represent an alternative way to solve the KKT system (1.7)–

(1.10), which consists in replacing the nonsmooth KKT condition (1.8) by a

smooth nonlinear approximation [16, 76]:

∇

L(z

)=0 (1.18)

0,i

, i = 1,2, ...,q (1.19)

≥ 0 (1.20)

g(z

) ≤ 0 , (1.21)

where

> 0 is a slack variable and g

) is the i-th constraint function. This

system is then solved with a Newton-type method. The obtained solution is not

a solution to the original NLP problem (1.1)–(1.2), but to the following problem

[16, 76, 8]:

∗

)=inf

[ f (z,x

B(z,x

)]. (1.22)

Here, B(z,x

) is the so called barrier function, which is nonnegative and contin-

uous over the region {z ∈R

|g(z,x

) < 0} and approaches ∞ as the boundary of

the feasible region {z ∈ R

|g(z,x

) ≤ 0} is approached from the interior. Thus,

the function B(z,x

) sets a barrier against leaving the feasible region. The solu-

tion of the barrier problem (1.22) requires for the optimization to start from a

point inside the region {z ∈R

|g(z,x

) < 0}. The IP methods are also referred to

as barrier function methods. They generate a sequence of feasible points whose

limit is an optimal solution to the original problem (1.1)–(1.2) [8]. If the optimal

solution occurs at the boundary of the feasible region, the procedure moves from

the interior to the boundary. Typically, the barrier function B(z,x

) has the form

[8]:

6 1 Multi-parametric Programming

B(z,x

∑

i=1

−1

(z,x

)

or B(z,x

)=−

∑

i=1

ln[−g

(z,x

)] . (1.23)

The solution of problem (1.22) is closer to the true solution the smaller

gets. An

important feature of the IP methods is that once a solution for a given

is found,

the parameter

can be reduced by a constant factor and an accurate solution

of the original NLP problem (1.1)–(1.2) is obtained after a limited number of

Newton iterations [16, 76]. The relation between the original problem (1.1)–(1.2)

and the barrier problem (1.22) is given by [8]:

∗

)= lim

→0

∗

)= inf

∗

) . (1.24)

1.1.3.2 Penalty Function Methods

Methods using penalty functions transform a constrained problem into a single un-

constrained problem or into a sequence of unconstrained problems [8]. The con-

straints are placed into the objective function via a penalty parameter in a way that

penalizes any violation of the constraints. The penalty function methods are also

referred to as the exterior penalty function methods, since they generate a sequence

of infeasible points whose limit is an optimal solution to the original problem [8].

Consider the problem (1.1)–(1.2) for a given x

∈ X. A penalty is desired only if

the point z is not feasible, i.e., if g(z,x

) > 0. A suitable unconstrained problem is

therefore given by [8]:

∗

)=inf

[ f (z,x

p(z,x

)] s.t. z ∈R

, (1.25)

where p(z,x

∑

i=1

[max{0, g

(z,x

)}]

is the so called penalty function, l ≥ 2is

an integer, and

> 0 is a penalty parameter. If g

(z,x

) ≤ 0, ∀i = 1,2,...,q then

max{0, g

(z,x

)} = 0, ∀i = 1,2, ...,q and no penalty is incurred, i.e., p(z,x

)=0.

On the other hand, if g

(z,x

) > 0, for some i,thenmax{0, g

(z,x

)} > 0andthe

penalty term

p(z,x

) is realized [8]. The condition l ≥ 2 ensures that the penalty

function p(z,x

) will be differentiable.

An important issue in the penalty function methods is the selection of the penalty

parameter

. Consider the penalty problem [8]:

∗

= sup

) , (1.26)

where W(

)=J

∗

). The relation between the primal problem (1.1)–(1.2) and

the penalty problem (1.26) is given by [8]:

∗

)=sup

)= lim

→∞

) (1.27)

From this result it is clear that we can get arbitrarily close to the optimal objective

value of the primal problem (1.1)–(1.2) by computing W (

) for a sufﬁciently large

1.1 Multi-parametric Nonlinear Programming 7

. However, as pointed out in [8], there are computational difﬁculties associated

with large penalty parameters, due to ill-conditioning. Therefore, most algorithms

using penalty functions solve a sequence of problems (1.25) for an increasing se-

quence of penalty parameters. With each new value of the penalty parameter, an

optimization technique is employed, starting with the optimal solution of prob-

lem (1.25) obtained for the parameter value chosen previously. Such an approach

is sometimes referred to as a sequential unconstrained minimization technique [8].

More details about the penalty function methods can be found in [8].

For a given

, the optimization problem (1.25) can be solved by applying the

steepest descent method [18]. Let h(z,x

)= f(z,x

p(z,x

). Then, the steep-

est descent direction from z is −∇

h(z,x

). With the method of steepest descent

[18], the values of optimization variables at the k +1-th iteration are obtained by the

formula:

k+1

= z

−

∇

h(z

) , (1.28)

where

> 0 is the step length. In order for the steepest descent method to be suc-

cessful, it is important to choose the step length

. One way to do this is to let

,where

∈ (0,1) and m ≥ 0 is the smallest nonnegative integer such that

there is a sufﬁcient decrease in h(z,x

).Thismeansthat:

h(z

−

∇

h(z

),x

) −h(z

) < −

∇

h(z

) , (1.29)

where

∈ (0,1). This strategy, introduced in [3], is an example of a line search in

which one searches on a ray from z

in a direction in which h(z,x

) is locally

decreasing. More details about the method of steepest descent can be found in [44].

Unfortunately, the methods based on steepest descent have slow local convergence,

even for very simple functions [44]. This is due to the fact that the steepest descent

direction scales with h(z, x

) and therefore the speed of convergence depends on

conditioning and scaling. A good alternative to the steepest descent method is the

conjugate gradient method [28, 1], which has improved local convergence proper-

ties. Also, the Newton-type methods can be successfully applied to solve the opti-

mization problem (1.25).

1.1.3.3 Direct Search Methods

The direct search methods do not use or approximate the objective function’s gradi-

ent, i.e. they represent derivative-free methods for optimization. These methods use

values of the objective function and constraints taken from a set of sample points

and use that information to continue the sampling. More precisely, the direct search

methods consist in a sequential examination of trial solutions involving comparison

of each trial solution with the best obtained up to that time together with a strategy

for determining (as a function of earlier results) what the next trial solution will be

[35]. There is a number of direct search methods for unconstrained optimization (see

for example [44, 51]). However, here, the most widely used direct search methods

for constrained nonlinear optimization are outlined.

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