# optimalControlReachsetMPC

solve an optimal control problem

## Contents

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

## Syntax

[param,res] = optimalControlReachsetMPC(x0,J,L,Opts)

## Description

This function solves the optimal control problem as defined in Eq. (7) in [1] to determine a feasible reference trajectory.

## Input Arguments

x0 | initial state of the considered reference trajectory part (dimension: [nx,1])
L | value of the objective function (=cost) of the previous solution
J | summed distance of the points from the previous solution to the terminal region
Opts | a structure containing following options
.funHandle | function handle to the dynamic function
.nx | number of system states
.nu | number of system inputs
.uMax | upper bound for the input constraints
.uMin | lower bound for the input constraints
.xf | goal state
.N | number of time steps for the prediction horizon. Prediction horizon: Opts.N * Opts.dT [{10} / positive integer]
.dT | time step. Prediction horizon: Opts.N * Opts.dT
.Q | state weighting matrix for the cost function (center trajectory)
.R | input weighting matrix for the cost function (center trajectory)
.termReg | terminal region around the steady state xf (class: mptPolytope)
.alpha | contraction rate for the contraction constraint [{0.1} / alpha > 0]
.maxIter | maximum number of iterations for the optimal control problem [{10} / positive integer]

## Output Arguments

param | a structure containing following options
.xc | reference trajectory (dimension: [nx,N+1])
.uc | reference trajectory control inputs (dimension: [nu,N])
.J | summed distance of the reference trajectory points from the terminal region
.L | value of the objective function (=cost) of the reference trajectory
res | flag that specifies if the computed reference trajectory is feasible (0 or 1)

## See Also

`optimalControl`, `reachsetMPC`

## References

- [1] Schuermann et al. (2018), Reachset Model Predictive Control for Disturbed Nonlinear Systems

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