# reduceRedundancies

Simplify system of first-order differential algebraic equations by eliminating redundant equations and variables

## Syntax

``````[newEqs,newVars] = reduceRedundancies(eqs,vars)``````
``````[newEqs,newVars,R] = reduceRedundancies(eqs,vars)``````

## Description

example

``````[newEqs,newVars] = reduceRedundancies(eqs,vars)``` eliminates redundant equations and variables from the system of first-order differential algebraic equations (DAEs) `eqs`. The input argument `vars` specifies the state variables of the system.`reduceRedundancies` returns the new DAE system as a column vector `newEqs` and the reduced state variables as a column vector `newVars`. Each element of `newEqs` represents an equation with right side equal to zero.```

example

``````[newEqs,newVars,R] = reduceRedundancies(eqs,vars)``` returns a structure array `R` containing information on the eliminated equations and variables.```

## Examples

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Simplify a system of five differential algebraic equations (DAEs) in four state variables to a system of two equations in two state variables.

Create the following system of five DAEs in four state variables `x1(t)`, `x2(t)`, `x3(t)`, and `x4(t)`. The system also contains symbolic parameters `a1`, `a2`, `a3`, `a4`, `b`, `c`, and the function `f(t)` that are not state variables.

```syms x1(t) x2(t) x3(t) x4(t) a1 a2 a3 a4 b c f(t) eqs = [a1*diff(x1(t),t)+a2*diff(x2(t),t) == b*x4(t), a3*diff(x2(t),t)+a4*diff(x3(t),t) == c*x4(t), x1(t) == 2*x2(t), x4(t) == f(t), f(t) == sin(t)]; vars = [x1(t),x2(t),x3(t),x4(t)];```

Use `reduceRedundancies` to eliminate redundant equations and corresponding state variables.

`[newEqs,newVars] = reduceRedundancies(eqs,vars)`
```newEqs =  ```
```newVars =  $\left(\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right)$```

Specify input order of the state variables to choose which variables are being returned when eliminating DAEs.

Create a system of four DAEs in four state variables `V_ac(t)`, `V1(t)`, `V2(t)`, and `I(t)`. The system also contains symbolic parameters `L`, `R`, and `V0`.

```syms V_ac(t) V1(t) V2(t) I(t) L R V0 eqs = [V_ac(t) == V1(t) + V2(t), V1(t) == I(t)*R, V2(t) == L*diff(I(t),t), V_ac(t) == V0*cos(t)]```
```eqs =  ```
`vars = [V_ac(t),I(t),V1(t),V2(t)]`
`vars = $\left(\begin{array}{cccc}{V}_{\mathrm{ac}}\left(t\right)& \text{I}\left(t\right)& {V}_{1}\left(t\right)& {V}_{2}\left(t\right)\end{array}\right)$`

Use `reduceRedundancies` to eliminate redundant equations and variables. `reduceRedundancies` prioritizes to keep the state variables in the vector `vars` starting from the first element.

`[newEqs,newVars] = reduceRedundancies(eqs,vars)`
```newEqs =  ```
`newVars = $\text{I}\left(t\right)$`

Here, `reduceRedundancies` returns a reduced equation in term of the variable `I(t)`.

When multiple ways of reducing the DAEs exist, specify a different input order of the state variables to choose which variables are being returned. Specify another vector that contains a different order of the state variables. Eliminate the DAEs again.

`vars2 = [V_ac(t),V1(t),V2(t),I(t)]`
`vars2 = $\left(\begin{array}{cccc}{V}_{\mathrm{ac}}\left(t\right)& {V}_{1}\left(t\right)& {V}_{2}\left(t\right)& \text{I}\left(t\right)\end{array}\right)$`
`[newEqs,newVars] = reduceRedundancies(eqs,vars2)`
```newEqs =  ```
`newVars = ${V}_{1}\left(t\right)$`

Here, `reduceRedundancies` returns a reduced equation in term of the state variable `V1(t)`.

Declare three output arguments when calling `reduceRedundancies` to simplify a system of equations and return information about the eliminated equations.

Create the following system of five differential algebraic equations (DAEs) in four state variables `x1(t)`, `x2(t)`, `x3(t)`, and `x4(t)`. The system also contains symbolic parameters `a1`, `a2`, `a3`, `a4`, `b`, `c`, and the function `f(t)` that are not state variables.

```syms x1(t) x2(t) x3(t) x4(t) a1 a2 a3 a4 b c f(t) eqs = [a1*diff(x1(t),t)+a2*diff(x2(t),t) == b*x4(t), a3*diff(x2(t),t)+a4*diff(x3(t),t) == c*x4(t), x1(t) == 2*x2(t), x4(t) == f(t), f(t) == sin(t)]; vars = [x1(t),x2(t),x3(t),x4(t)];```

Call `reduceRedundancies` with three output arguments.

`[newEqs,newVars,R] = reduceRedundancies(eqs,vars)`
```newEqs =  ```
```newVars =  $\left(\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right)$```
```R = struct with fields: solvedEquations: [2x1 sym] constantVariables: [x4(t) f(t)] replacedVariables: [x2(t) x1(t)/2] otherEquations: f(t) - sin(t) ```

The function `reduceRedundancies` returns information about eliminated equations to `R`. Here, `R` is a structure array with four fields.

The `solvedEquations` field contains the equations that are eliminated by `reduceRedundancies`. The eliminated equations contain those state variables from `vars` that do not appear in `newEqs`. The right side of each eliminated equation is equal to zero.

`R1 = R.solvedEquations`
```R1 =  $\left(\begin{array}{c}{x}_{1}\left(t\right)-2 {x}_{2}\left(t\right)\\ {x}_{4}\left(t\right)-f\left(t\right)\end{array}\right)$```

The `constantVariables` field contains a matrix with two columns. The first column contains those state variables from `vars` that `reduceRedundancies` replaced by constant values. The second column contains the corresponding constant values.

`R2 = R.constantVariables`
`R2 = $\left(\begin{array}{cc}{x}_{4}\left(t\right)& f\left(t\right)\end{array}\right)$`

The `replacedVariables` field contains a matrix with two columns. The first column contains those state variables from `vars` that `reduceRedundancies` replaced by expressions in terms of other variables. The second column contains the corresponding values of the eliminated variables.

`R3 = R.replacedVariables`
```R3 =  $\left(\begin{array}{cc}{x}_{2}\left(t\right)& \frac{{x}_{1}\left(t\right)}{2}\end{array}\right)$```

The `otherEquations` field contains those equations from `eqs` that do not contain any of the state variables `vars`.

`R4 = R.otherEquations`
`R4 = $f\left(t\right)-\mathrm{sin}\left(t\right)$`

## Input Arguments

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System of first-order DAEs, specified as a vector of symbolic equations or expressions.

The relation operator `==` defines symbolic equations. If you specify the element of `eqs` as a symbolic expression without a right side, then a symbolic equation with right side equal to zero is assumed.

State variables, specified as a vector of symbolic functions or function calls, such as `x(t)`.

The input order of the state variables determines which reduced variables are being returned. If multiple ways of reducing the DAEs exist, then `reduceRedundancies` prioritizes to keep the state variables in `vars` starting from the first element.

Example: `[x(t),z(t),y(t)]`

## Output Arguments

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System of first-order DAEs, returned as a column vector of symbolic expressions. Each element of `newEqs` represents an equation with right side equal to zero.

Reduced set of variables, returned as a column vector of symbolic function calls.

Information about eliminated variables, returned as a structure array containing four fields. To access this information, use:

• `R.solvedEquations` to return a symbolic column vector of all equations that `reduceRedundancies` used to replace those state variables that do not appear in `newEqs`.

• `R.constantVariables` to return a matrix with the following two columns. The first column contains those original state variables of the vector `vars` that were eliminated and replaced by constant values. The second column contains the corresponding constant values.

• `R.replacedVariables` to return a matrix with the following two columns. The first column contains those original state variables of the vector `vars` that were eliminated and replaced in terms of other variables. The second column contains the corresponding values of the eliminated variables.

• `R.otherEquations` to return a column vector containing all original equations `eqs` that do not contain any of the input variables `vars`.