GPOPS2用户手册：多相最优控制问题的MATLAB解决方案

需积分: 50 75 浏览量更新于2024-07-20 2 收藏 1.32MB PDF 举报

GPOPS2用户手册是一份详尽的指南，针对的是Version 2.3版本的通用MATLAB优化控制软件。该软件专为解决多种阶段的最优控制问题而设计，利用了可变阶次自适应正交列化方法（Variable-Order Adaptive Orthogonal Collocation Method）以及稀疏非线性规划技术。GPOPS2的适用范围广泛，涵盖工程、经济和医学等多个领域的应用。核心功能上，GPOPS2集成了最新的正交列化算法进展，尤其是采用了可变阶次的Legendre-Gauss-Radau积分插值法。这种技术允许软件在求解过程中动态调整节点分布，提高精度和效率。值得注意的是，GPOPS2与两个强大的非线性优化器紧密集成：SNOPT（用户需自行获取）和IPOPT，后者提供了MATLAB Mex文件，使得用户可以直接在其环境中利用IPOPT的功能。手册详细介绍了如何使用GPOPS2进行建模、设置参数和调优控制过程，包括如何处理非线性约束和目标函数，以及如何有效地处理大规模问题。此外，它还包含了示例案例和故障排除指南，帮助用户快速理解和掌握软件操作。对于新用户来说，这份用户手册是入门GPOPS2的宝贵资源，对于经验丰富的优化工程师，它则是一个提升现有技能和探索新方法的强大工具。无论是初次接触还是深入研究，都能从中找到所需的信息和策略来优化复杂的控制系统。版权方面，所有权利归Michael A. Patterson和Anil V. Rao所有，确保了技术的独占性和质量保证。

2.4 Syntax of Endpoint Function setup.functions.endpoint 12

• bounds.eventgroup(g).lower and bou nd s. eventgroup(g).upper: row vectors of length n

(g)

that

contain the lower and upper bounds on the group g = 1, . . . , G of event constraints. The row vectors

bounds.eventgroup(g).lower and bounds. eventgroup(g).upper have the following form:

bounds.eventgroup(g).lower =

lower

··· b

lower

(g)

bounds.eventgroup(g).upper =

upper

··· b

upper

(g)

Note: any ﬁe ld s that do not apply to a pr obl em (for example, a problem with no path const r aints) should

be omitted completely.

2.4 Syntax of Endpoint Function

setup.functions.endpoint

The syntax used to evaluate the user-deﬁned endpoint function deﬁne d by the function handle setup.functions.endpoint

is given as follows:

function output=endpointfun(input)

The input input is a structure that contains the ﬁ eld s phase, auxdata, and parameter if the problem has

parameters. The ﬁeld

input.phase is an array of structures of length P (where P is the number of phases)

such that the p

element of

input.phase contains the following ﬁelds:

• input.phase(p).initialtime: a scalar that contains the initial time in phase p = 1, . . . , P;

•

input.phase(p).ﬁnaltime: a scalar that contains the ﬁnal time in phase p = 1, . . . , P ;

• input.phase(p).initialstate: a row vector of length n

(p)

that contains the initial state in phase p =

1, . . . , P ;

•

input.phase(p).ﬁnalstate: a row vector of length n

(p)

that contains the ﬁnal state in phase p =

1, . . . , P ;

•

input.phase(p).integral: a row vector of length n

(p)

that contains the integrals in phase p = 1, . . . , P ;

•

input.parameter: a row vector of length n

that contains the static parameters in phase p = 1, . . . , P ;

The ﬁeld input.auxd ata contains the same information as the ﬁeld input.auxdata that was speciﬁed in

the structure input that was used to specify the information for the entire problem. The output output

is a structure that contains the ﬁelds objective and eventgroup. The ﬁelds output.objective and out-

put.eventgroup are given as follows:

• output.objective: a scalar that contains the result of computing the objective function on the current

call to

input.functions.endpoint;

•

output.eventgroup: an array of structures of length G (whe r e G is th e number of event groups)

such that the g

element in

output.eventgroup is a row vector of le ngth n

(g)

that contains the

result of evaluating g

group of event constraints at the values given in the call t o the function

input.functions.endpoint;

2.5 Syntax for Continuous Function

setup.functions.continuous

The syntax used to evaluate the continuous functions deﬁned by the functi on handle setup.functions.continuous

is given as follows:

function output=continuousfun(input)

The input

input is a structure that contains the ﬁelds phase and auxdata. The ﬁeld in p ut .p has e is an

array of structures of length P (where P is the number of phases) such that the p

element of input.phase

contains the following ﬁelds:

2.6 Specifying an Initial Guess of The Solution 13

• input.phase(p).time: a column vector of length N

(p)

, where N

(p)

is the number of collocation points

in phase p = 1, . . . , P .

•

input.phase(p).state: a matrix of size N

(p)

×n

(p)

, where N

(p)

and n

(p)

are, respectively, the number

of colloc ation points and the dimension of the state in phase p = 1, . . . , P;

•

input.phase(p).control: a matrix of size N

(p)

×n

(p)

, where N

(p)

and n

(p)

are, respectively, the number

of colloc ation points and the dimension of the control in phase p = 1, . . . , P ;

•

input.phase(p).parameter: a matrix of size N

(p)

×n

, where N

(p)

is the number of collocation points

in phase p = 1, . . . , P and and n

is the dimension of the static parameter. [Note: see below for the

reason why the static parameter has a size N

(p)

× n

];

Finally, output is an array of structures of length P (where P is the number of phases) such that the p

element of

output contains the following ﬁelds:

•

output.dynamics: a matrix of size N

(p)

× n

(p)

, where N

(p)

and n

(p)

are, respectively, the number of

collocation points and the dimension of th e state in phase p = 1, . . . , P ;

•

output.path: a matrix of s ize N

(p)

× n

(p)

, where N

(p)

and n

(p)

are, respectively, the numb e r of

collocation points and the number of path constraints in phase p = 1, . . . , P ;

•

output.integrand: a matri x of s i ze N

(p)

× n

(p)

, where N

(p)

and n

(p)

are, respectively, the number of

collocation points and the number of integrals in phase p = 1, . . . , P ;

IMPORTANT NOTE: While i t may seem a bit odd, th e ﬁeld

input.phase(p).parameter is actually

speciﬁed as if it were phase-dependent while it actually does not depend upon the p hase because the static

parameters themselves are independ ent of the phase. Furthermore, while the static parameters are deﬁned

as a single row vector, the arrays

input.phase(p).parameter are actually matrices of size N

(p)

×n

, where

(p)

is the number of collocation points in each phase. The reason for making the st atic parameters phase

dependent and providing them as an array with N

(p)

rows is to improve the eﬃciency with which the NLP

derivatives are computed.

2.6 Specifying an Initial Guess of The Solution

The ﬁeld guess of the user-deﬁned structure setup contains the initial guess for the problem. The ﬁeld guess

is a then struc tu r e that contains the ﬁelds phase and parameter. Assume th at M

(p)

is the number of values

used in the guess for the time, state, and control in phase p = 1, . . . , P. The ﬁeld

setup.guess.phase is an

array of structures of length P such that the p

element of setup.guess.phase contains the following ﬁelds:

•

setup.guess.phase(p).time: a column vector of length M

(p)

in phase p = 1, . . . , P ;

•

setup.guess.phase(p).state: a matrix of size M

(p)

× n

(p)

, wher e n

(p)

is the dimension of the state in

phase p = 1, . . . , P ;

• setup.guess.phase(p).control: a matrix of size M

(p)

×n

(p)

, where n

(p)

is the dimension of the control

in phase p = 1, . . . , P ;

• setup.guess.phase(p).integral: a row vector of length n

(p)

, wh er e n

(p)

is the number of integrals in

phase p = 1, . . . , P ;

•

setup.guess.parameter: a row vector of length size n

, where n

is the number of static parameters

in the pr oble m.

It is noted that the column vector of time points speciﬁed in each phase p = 1, . . . , P in the ﬁeld

setup.guess.phase(p).time

must be monotonically increasing.

2.7 Use of ADiGator

to Generate NLP Solver Derivatives 14

2.7 Use of ADiGator

to Generate NLP Solver Derivatives

As stated earlier, one of the options for gener atin g derivatives required by the NLP solver is to use the open-

source automatic MATLAB diﬀerentiation software ADiGator.

The software ADiGator is described in detail

in Ref.

1 and can be found at http://sourceforge.net/projects/adigator. The option for using ADiGa-

tor is invoked by setting setup.derivatives.method = ’adigator’. Assuming the user has obtained ADiGator

from http://sourceforge.net/projects/adigator and the option setup.derivatives.method has been

set to ’adigator], the ﬁrst and/or second derivative functions (depending upon which NLP solver is being used

an which derivative level is chosen) of the user-supplied optimal control functions

setup.functions.continuous

setup.functions.endpoint are obtained simply by executing GPOPS −II on the user setu p. At the start of

the GPOPS − II run, screen output will be displayed that provides a status of the derivative ﬁle generation.

After generating the derivative ﬁles, GPOPS − II will run in the usual mode, this time using the derivative

code generated by ADiGator instead of using the default sparse ﬁnite-diﬀerence method.

An important aspect of using ADiGator is that the us e r may only want to generate derivative cod e once

and then re-use the derivative funct ions in future runs of GPOPS − II. In order to avoid regeneration of the

user

continuous and endpoint functions, it is necessary that the user provide these derivative function

names in the setup structure that will be used in the f ut ur e execution of GPOPS − II. First, let myproblem be

the preﬁx for the continuous and endpoint func tion names (that is, the continuous and endpoint functions

are m-ﬁles named

myproblemContinuous and myproblemEndpoint). The ﬁrst derivative functions are

then m-ﬁles named myproblemContinuousADiGatorGrd and myproblemEndpointADiGatorGrd.m

while the second derivative functions are m-ﬁles named myproblemContinuousADiGatorHes and myprob-

lem

EndpointADiGatorHes. Corresponding to the ﬁrst and second derivative function names created by

ADiGator, the following ﬁelds must be added to the setup structure used in a fut ur e run of GPOPS − II:

•

setup.adigatorgrd.continuous = @myproblemContinuousADiGatorGrd

• setup.adigatorgrd.endpoint = @myproblemEndpointADiGatorGrd

• setup.adigatorhes.continuous = @myproblemContinuousADiGatorHes

• setup.adigatorhes.endpoint = @myproblemEndpointADiGatorHes

Finally, a few important points should be noted when using ADiGator with GPOPS − II. The ﬁrst important

point pertains t o when the derivative ﬁles are regenerated. Speciﬁcally, it is recommended to regenerate the

derivative ﬁles if the user has any doubt as to whether any changes have been made to anyth ing in the

problem setup or the associated function ﬁles. For example, if the user continuous or endpoint function

depends upon another function and the user has made changes in the dependent function (but not in the

actual user functions themselves), then it will be necessary to regenerate the derivative functions using

ADiGator. The easi es t way to know that the correct derivative ﬁles are used in subsequent executions of

GPOPS − II is to delete any or all of the ﬁles create d by ADiGator before executing GPOPS − II using th e

derivative option

setup.derivatives.method = ’adigator’.

A second important point regarding ADiGator pertains to the situation when GPOPS − II is unable

to generate derivative ﬁles. As with any automatic diﬀerentiation software, ADiGator has limitations on

function co de that can be diﬀerentiated. If a GPOPS −II user function cannot be diﬀerentiated by ADiGator,

GPOPS − II will produce the following generic error:

GPOPS-II ERROR: ADiGator could not produce derivative ﬁles

Note that the above error does not provide the reason why ADiGator cannot produce derivative ﬁles. In

order for the user to ascertain the reason why ADiGator cannot produce derivative ﬁles, it is necessary to

run the function

adigatorGenFiles4gpops2 on the user setu p structure. In fact, derivative ﬁles can be

generated without running GPOPS − II simply by executing the command

adigatorﬁlenames = adigatorGenFiles4gpops2(setup)

It is also noted that ADiGator is upd ated regularly when bugs are found. For more information about using

ADiGator, the user is referred to the ADiGator user’s guide as found on on the ADiGator project page

http://sourceforge.net/projects/adigator.

2.8 Scaling the Optimal Control Problem 15

2.8 Scaling the Optimal Control Problem

It is always preferable for the user to scale an optimal control problem of interest based on an understand in g

of th e probl em itsel f. In many cases, however, it may be diﬃcult to determine a suitable scaling of the

optimal control problem. While it is beyond the scope of the software to provide a general procedure for

scaling, several diﬀerent scaling options have been provided in GPOPS − II that may prove usef ul to the user

in lieu of manually scaling the problem. We now explain these options in more detail than were ex pl aine d

earlier in this us e r ’s guide.

Setting

setup.scales.method to ’automatic-bounds’ scales the problems based on the variable bounds

and by computing scale factors based on randomly sampling the ﬁrst derivatives of the optimal control

problem functions.

Setting setup.scales.method to ’automatic-guess’ scales the problem once us in g the

initial guess supplied by the user. Note that the option ’automatic-guess’ does not re-scale the problem on

each mesh reﬁnement iteration, but only scales the problem a single time based purely on whatever guess

the user provides. Setting

setup.scales.method to ’automatic-guessUpdate’ scales the problem on the user-

supplied initial guess on the ﬁrst mesh and from the solution generated by GPOPS −II on every mesh during

the mesh reﬁnement. As a result, the option ’automatic-guessUpdate’ performs a new scaling of the optimal

control problem for every mesh on which the problem is solved. Setting

setup.scales.method to ’automatic-

hybrid’ provides a hybrid scaling that uses the same method as ’automatic-bounds’ on the ﬁrst me sh and

then performs a single scaling from the solution obtained on the ﬁrst mesh. Setting

setup.scales.method to

’automatic-hybridUpdate’ scales the problem using the method ’automatic-bounds’ on the initial mesh and

using the method ’automatic-guessUpdate’ on every subsequent mesh during the mesh reﬁnement. In orde r

to maximize repeatability,

scales.method is changed to ’deﬁned’ when using any scaling method where

the sc ales are not changed between mesh iterations (in other words, the scales are set to ’deﬁned’ when

using ’automatic-bounds’, ’automatic-gues s ’, or ’automatic-hybrid’). Finally, other ﬁelds in the structure

setup.scales are created during an execut ion of GPOPS − II but the user need not in gen er al be concerned

with these add it ional ﬁelds. If a user desir es an explanation of these other ﬁelds th at are created, please

contact the authors.

3 Output from an Execution of GPOPS − II

The output of an execution of GPOPS − II is the structure

output, where output contains the following

ﬁelds:

•

result: a structure that contains the following ﬁelds:

–

solution: the optimal time, state, and control, in each phase and the optimal value of the static

parameter vector. The optimal time, state, and control are stored, respectively, in the ﬁelds

solution.phase(p).time, sol u tion .p ha se(p). stat e, and solution.phase(p).control, while th e

static parameter is st or ed in the ﬁeld solution.parameter;

–

objective: the optimal value of the objective function of the optimal control pr oble m;

•

result.setup: the setup structure that produced the result found in result with GPOPS − II;

•

result.nextsetup: the setup structure that would be used for the next mesh reﬁnement iteration

had the mesh reﬁnement process not terminated on the previ ous mesh (due to the mesh reﬁnement

accuracy tolerance not having been satisﬁed).

•

meshhistory: the solution and error estimate for each mesh on which the NLP was solved (only if

mesh reﬁnement is used);

•

meshiterations: the number of mesh reﬁnement iter ati ons that were taken by GPOPS − II (only if

mesh reﬁnement is used);

剩余70页未读，继续阅读

atmkxx

粉丝: 1
资源: 1

GPOPS2用户手册：多相最优控制问题的MATLAB解决方案

GPOPS2软件包、使用说明、安装方式等

gpops工具箱+例子（详细注释）+官方手册+安装方法

gpops2UsersGuide.rar_GPOPS_gpops2userguide_gpops使用方法_gpops手册_最优控

GPOPS II include usersGuide_hp自适应_Raoph_matlab伪谱法_伪谱法_高斯伪谱法_

CarMaker的用户指导手册UsersGuide

QAC UsersGuide 用户使用手册

WebMethods Developer UsersGuide用户入门操作手册.pdf

ｍａｔｌａｂ工具箱GPOPS2.rar

MatlabFilterDesignHDLCoder2UsersGuide-hdlfilter.pdf

Mercury UsersGuide Mercury UsersGuide Mercury UsersGuide

最新资源