CasADi 3.4.4 用户指南：符号计算与优化工具

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CasADi是一款强大的数值优化工具，专用于符号代数、优化计算和自动微分，特别适合于构建高效的数学程序。该文档是CasADi v3.4.4的用户手册，由Joel Andersson、Joris Gillis和Moritz Diehl编写，发布日期为2018年5月19日。以下是文档的主要章节概览： 1. **简介** - CasADi是一个用于快速开发高效数学程序的工具包，支持符号计算、优化问题求解和算法自动微分。 - 介绍CasADi的功能特点，如非线性优化、符号表达式处理、函数对象管理和动态系统建模。 - 提供帮助和支持渠道，以及引用该软件时的注意事项。 2. **获取和安装** - 介绍了CasADi的下载途径和安装步骤，确保用户能够在本地环境中顺利运行。 3. **符号框架** - 分别讲解了SX（符号表达式）和MX（多行向量）两种核心符号数据类型。 - 提供了混合使用SX和MX、矩阵操作（如索引设置、算术运算）、查询属性以及线性代数和微分算法的指导。 4. **函数对象** - 解释了如何调用函数对象、MX转换为SX、处理非线性根查找问题、初始值问题和敏感性分析。 - 包括创建和使用数值积分器、优化问题求解器（包括非线性规划和二次规划）以及循环结构的等价实现（如map和fold）。 5. **生成C代码** - 重点讲述了如何利用CasADi的generate功能将函数子集转换为C代码，以加速数值评估速度。 - 提供了生成C代码的主要API和使用方法，这对于需要高性能计算的场景尤其重要。 6. **用户自定义函数** - 用户可以根据需求创建自定义函数，结合CasADi的符号计算和优化能力，实现复杂的数学模型。 7. **DaeBuilder类** - 专门用于构建复杂动力学模型，辅助最优控制算法的设计与实现。 8. **最优控制问题求解** - 详细介绍如何在CasADi中处理最优控制问题，涉及决策变量、参数、目标函数、约束条件和初始条件的管理。 9. **opti堆栈集成** - 强调了opti类的帮助，它整合了决策变量、参数、目标函数和约束等核心组件，便于构建和求解优化问题。 CasADi文档详细阐述了如何利用其强大的符号计算和优化功能，解决实际工程中的数学问题，提供了丰富的工具和接口，使得用户能够方便地进行符号代数建模、求解优化问题以及数值评估的性能优化。无论是科研还是工业应用，都能从中找到所需的信息和技术支持。

3.1. THE SX SYMBOLICS 13

• SX.eye(n): Create an n-by-n diagonal matrix with ones on the diagonal and structural

zeros elsewhere.

• SX(scalar type): Create a scalar (1-by-1 matrix) with value given by the argument.

This method can be used explicitly, e.g. SX(9), or implicitly, e.g. 9 ∗ SX.ones(2,2).

• SX(matrix type): Create a matrix given a numerical matrix given as a NumPy or

SciPy matrix (in Python) or as a dense or sparse matrix (in MATLAB/Octave). In

MATLAB/Octave e.g. SX ([1,2,3,4]) for a row vector, SX ([1;2;3;4]) for a column

vector and SX ([1,2;3,4]) for a 2-by-2 matrix. This method can be used explicitly or

implicitly.

• repmat(v,n,m): Repeat expression v n times vertically and m times horizontally.

repmat(SX(3),2,1) will create a 2-by-1 matrix with all elements 3.

• (Python only) SX(list): Create a column vector (n-by-1 matrix) with the elements

in the list, e.g. SX ([1,2,3,4]) (note the diﬀerence between Python lists and MAT-

LAB/Octave horizontal concatenation, which both uses square bracket syntax)

• (Python only) SX(list of list ): Create a dense matrix with the elements in the lists,

e.g. SX ([[1,2],[3,4]]) or a row vector (1-by-n matrix) using SX ([[1,2,3,4]]) .

Note for MATLAB/Octave users

In MATLAB, if the import command is omitted, you can still use CasADi by preﬁxing all

the symbols with the package name, e.g. casadi.SX instead of SX, provided the casadi

package is in the path. We will not do this in the following for typographical reasons, but

note that it is often preferable in user code. In Python, this usage corresponds to issuing

”import casadi” instead of ”from casadi import ∗”.

Unfortunately, Octave (version 4.0.3) does not implement MATLAB’s import com-

mand. To work around this issue, we provide a simple function import.m that can be

placed in Octave’s path enabling the compact syntax used in this guide.

Note for C++ users

In C++, all public symbols are deﬁned in the casadi namespace and require the inclusion

of the casadi/casadi.hpp header ﬁle. The commands above would be equivalent to:

// C++

#include <c a s a d i / ca s a d i . hpp>

using namespace ca s a d i ;

int main () {

SX x = SX : : sym( ”x” ) ;

SX y = SX : : sym( ”y” , 5 ) ;

SX Z = SX : : sym ( ”Z” , 4 , 2 )

3.3. THE MX SYMBOLICS 15

@1=3,

[[((@1*x_0)+y), ((@1*x_2)+y)],

[((@1*x_1)+y), ((@1*x_3)+y)]]

(2, 2)

As you can see, the output of this operation is a 2-by-2 matrix. Note how the multi-

plication and the addition were performed element-wise and new expressions (of type SX)

were created for each entry of the result matrix.

We shall now introduce a second, more general matrix expression type MX. The MX type

allows, like SX, to build up expressions consisting of a sequence of elementary operations.

But unlike SX, these elementary operations are not restricted to be scalar unary or binary

operations (R → R or R × R → R. Instead, the elementary operations that are used to

form MX expressions are allowed to be general multiple sparse-matrix valued input, multiple

sparse-matrix valued output functions: R

×m

× . . . × R

×m

→ R

×q

× . . . × R

×q

The syntax of MX mirrors that of SX:

# Python

x = MX. sym( ’ x ’ , 2 , 2 )

y = MX. sym( ’ y ’ )

f = 3∗x + y

print ( f )

print ( f . shape )

% MATLAB/ Octave

x = MX. sym( ’ x ’ , 2 , 2 ) ;

y = MX. sym( ’ y ’ ) ;

f = 3∗x + y ;

disp ( f )

disp ( s iz e ( f ) )

((3*x)+y)

(2, 2)

Note how the result consists of only two operations (one multiplication and one ad-

dition) using MX symbolics, whereas the SX equivalent has eight (two for each element of

the resulting matrix). As a consequence, MX can be more economical when working with

operations that are naturally vector or matrix valued with many elements. As we shall see

in Chapter 4, it is also much more general since we allow calls to arbitrary functions that

cannot be expanded in terms of elementary operations.

MX supports getting and setting elements, using the same syntax as SX, but the way it

is implemented is very diﬀerent. Test, for example, to print the element in the upper-left

corner of a 2-by-2 symbolic variable:

# Python

x = MX. sym( ’ x ’ , 2 , 2 )

print ( x [ 0 , 0 ] )

% MATLAB/ Octave

x = MX. sym( ’ x ’ , 2 , 2 ) ;

x ( 1 , 1 )

x[0]

The output should be understood as an expression that is equal to the ﬁrst (i.e. index

0 in C++) structurally non-zero element of x, unlike x 0 in the SX case above, which is

the name of a symbolic primitive in the ﬁrst (index 0) location of the matrix.

Similar results can be expected when trying to set elements:

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casadi：CasADi是用于数值优化的符号框架，可在稀疏矩阵值计算图上以正向和反向模式实现自动微分。它支持自包含的C代码生成，并接口最新的代码，例如SUNDIALS，IPOPT等。它可以在C ++，Python或MatlabOctave中使用