# computeTermReglinSys

compute terminal region with linear system approach, using only containment

## Contents

- Syntax
- Description
- Input Arguments
- Output Arguments
- See Also
- References

## Syntax

T = computeTermReglinSys(benchmark, Param, Opts)

## Description

This function computes a terminal region with the approach in [1] and [2].

## Input Arguments

benchmark | name of the considered benchmark model (see; "aroc/benchmarks/dynamics/...")
Param | a structure containing the benchmark parameters
.U | set of admissible control inputs
.W | set of uncertain disturbances
.V | set of measurement noise
.X | set of state constraints
.Y | set of output constraints
.Yexact | set of error-free output constraints
.F -.gamma | defines the output equation y = x + Fv + gamma
U, X, and Y need to be polytopes, or will be transformed to polytopes within the algorithm (i.e., intervals or zonotopes; can also be used here, but will be transformed to polytopes). W, V, and Ustart on the other hand need to be intervals or zonotopes.
Opts | a structure containing the algorithm settings
.N | number of time steps [{10} / positive integer]
.timeStep | sampling time [{0.1} / positive scalar]
.terminalRegionType | method used for computing the terminal region [{`'implicit'`} or `'fixed'`]
.terminalRegionSet | set type of the terminal region [{`'zonotope'`} or `'ellipsoid'`]
.taylorOrder | Taylor order for certain approximations needed during the reachability analysis [{5} / positive integer]
.genMethod | method for computing the fixed directions [{`'spherical'`} or `'provided'`]
.nGenerators | if .genMethod=`'spherical'`, .nGenerators should be the number of generators for the fixed generator matrix [{dim(X)}/ positive integer]
.G | if .genMethod=`'provided'`, .matrix should be the desired matrix for the fixed generators
.K | feedback matrix for the terminal controller
.Q | state weighting matrix for the LQR approach applied to determine the terminal controller [{eye(nx)} / positive definite; square matrix]
.R | input weighting matrix for the LQR approach applied to determine the terminal controller [{eye(nu)} / positive definite; square matrix]

## Output Arguments

T | object of class termRegContainmentLinSys

## See Also

`terminalRegion`, termRegLinSysApproach

## References

- *[1] "Approximability of the Containment Problem for Zonotopes and Ellipsotopes", A. Kulmburg et al., submited to TAC in 2024
- [2] Gruber et al. (2021), Computing safe sets of linear sampled-data systems, IEEE Control Syst. Lett., vol. 5, no. 2, pp. 385-390

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