# equations

Define component equations

## Syntax

equations==`Expression1`

; end`Expression2`

## Description

`equations`

begins the equation section in a component file; this section
is terminated by an `end`

keyword. The purpose of the equation section is to
establish the mathematical relationships among a component’s variables, parameters, inputs,
outputs, time and the time derivatives of each of these entities. All members declared in the
component are available by their name in the equation section.

The equation section of a Simscape™ file is executed throughout the simulation. You can also specify equations that
are executed during model initialization only, by using the `(Initial=true)`

attribute. For more information, see Initial Equations.

The following syntax defines a simple equation.

equationsExpression1==Expression2; end

The statement

is an equation statement. It specifies
continuous mathematical equality between two objects of class * Expression1* ==

`Expression2`

`Expression`

.
An `Expression`

is a valid MATLAB^{®}expression.

`Expression`

may be constructed from any of the
identifiers defined in the model declaration.The equation section may contain multiple equation statements. You can also specify
conditional equations by using `if`

statements as follows:

equations if Expression EquationList { elseif Expression EquationList } else EquationList end end

**Note**

The total number of equation expressions, their dimensionality, and their order must be
the same for every branch of the `if-elseif-else`

statement.

You can declare intermediate terms in the `intermediates`

section of a component or domain file and then use these
terms in any equations section in the same component file, in an enclosing composite
component, or in a component that has nodes of that domain type.

You can also define intermediate terms directly in equations by using
`let`

statements as follows:

equations letdeclaration clauseinexpression clauseend end

The declaration clause assigns an identifier, or set of identifiers, on the left-hand side
of the equal sign (`=`

) to an equation expression on the right-hand side of
the equal sign:

LetValue= EquationExpression

The expression clause defines the scope of the substitution. It starts with the keyword
`in`

, and may contain one or more equation expressions. All the expressions
assigned to the identifiers in the declaration clause are substituted into the equations in
the expression clause during parsing.

**Note**

The `end`

keyword is required at the end of a
`let-in-end`

statement.

The following rules apply to the equation section:

`EquationList`

is one or more objects of class`EquationExpression`

, separated by a comma, semicolon, or newline.

`EquationExpression`

can be one of:`Expression`

Conditional expression (

`if-elseif-else`

statement)Let expression (

`let-in-end`

statement)

`Expression`

is any valid MATLAB expression. It may be formed with the following operators:Arithmetic

Relational (with restrictions, see Use of Relational Operators in Equations)

Logical

Primitive Math

Indexing

Concatenation

In the equation section,

`Expression`

may not be formed with the following operators:Matrix Inversion

MATLAB functions not listed in Supported Functions

The

`colon`

operator may take only constants or`end`

as its operands.All members of the component are accessible in the equation section, but none are writable.

The following MATLAB functions can be used in the equation section. The table contains additional restrictions that pertain only to the equation section. It also indicates whether a function is discontinuous. If the function is discontinuous, it introduces a zero-crossing when used with one or more continuous operands.

All arguments that specify size or dimension must be unitless constants or unitless compile-time parameters.

**Supported Functions**

Name | Restrictions | Discontinuous |
---|---|---|

`ones` | ||

`zeros` | ||

`cat` | ||

`horzcat` | ||

`vertcat` | ||

`length` | ||

`ndims` | ||

`numel` | ||

`size` | ||

`isempty` | ||

`isequal` | Possibly, if arguments are real and have the same size and commensurate units | |

`isinf` | Yes | |

`isfinite` | Yes | |

`isnan` | Yes | |

`plus` | ||

`uplus` | ||

`minus` | ||

`uminus` | ||

`mtimes` | ||

`times` | ||

`mpower` | ||

`power` | ||

`mldivide` | First argument must be a scalar | |

`mrdivide` | Second argument must be a scalar | |

`ldivide` | ||

`rdivide` | ||

`mod` | Yes | |

`sum` | ||

`prod` | ||

`floor` | Yes | |

`ceil` | Yes | |

`fix` | Yes | |

`round` | Yes | |

`eq` | Do not use with continuous variables | |

`ne` | Do not use with continuous variables | |

`lt` | ||

`gt` | ||

`le` | ||

`ge` | ||

`and` | Yes | |

`or` | Yes | |

`logical` | Yes | |

`sin` | ||

`cos` | ||

`tan` | ||

`asin` | ||

`acos` | ||

`atan` | ||

`atan2` | Yes | |

`log` | ||

`log10` | ||

`sinh` | ||

`cosh` | ||

`tanh` | ||

`exp` | ||

`sqrt` | ||

`abs` | Yes | |

`sign` | Yes | |

`any` | Yes | |

`all` | Yes | |

`min` | Yes | |

`max` | Yes | |

`double` | ||

`int32` | Yes | |

`uint32` | Yes | |

`repmat` | ||

`reshape` | Expanded empty dimension is not supported | |

`dot` | ||

`cross` | ||

`diff` | In the two argument overload, the upper bound on the second argument is 4, due to a Simscape limitation |

### Initial Equations

The `(Initial=true)`

attribute lets you specify equations that are
executed during model initialization only:

equations (Initial=true)Expression1==Expression2; end

The default value of the `Initial`

attribute for equations is
`false`

, therefore you can omit this attribute when declaring regular
equations.

For more information on when and how to specify initial equations, see Initial Equations.

## Examples

For a component where *x* and *y* are
declared as 1x1 variables, specify an equation of the form *y* = *x*^{2}:

equations y == x^2; end

For the same component, specify the following piecewise equation:

$$y=\{\begin{array}{ll}x\hfill & \text{for}-1=\text{}x=1\hfill \\ {x}^{2}\hfill & \text{otherwise}\hfill \end{array}$$

This equation, written in the Simscape language, would look like:

equations if x >= -1 && x <= 1 y == x; else y == x^2; end end

If a function has multiple return values, use it in a `let`

statement
to access its values. For example:

equations let [m, i] = min(a); in x == m; y == i; end end

## See Also

`assert`

| `delay`

| `der`

| `function`

| `integ`

| `intermediates`

| `tablelookup`

| `time`

**Introduced in R2009a**