# initialGuessSet_RiccatiEq

solves the Riccati equation for the linearized dynamics and encloses the resulting ellipsoid by a zonotope.

## Contents

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

## Syntax

PIset = initialGuessSet_RiccatiEq(obj,dynamics,Q,R)
PIset = initialGuessSet_RiccatiEq(obj,dynamics,Q,R,flag_invariant)

## Description

To obtain an initial guess for the optimization-based approaches for computing terminal regions of nonlinear systems, we solve the algebraic Riccati equation (ARE) for the linearized dynamics. The positively invariant (PI) ellipsoid whose shape matrix is the solution of the ARE is enclosed by a zonotope. By default, the order of the PI set is et to a user-defined value (obj.termRegOrder). If the additional input argument flag_invariant is set to true, the order of the zonotope enclosing the PI set is increased until we obtain an invariant outer approximation of the PI set (inspired by [2, Alg. 2, line 10-16]).

## Input Arguments

obj | instance of class computeTermRegNonlinSysOptApproach
dynamics | instance of class termRegNonlinSysDynamics
Q | state weighting matrix (positive-definite matrix of dimension; [obj.nx,obj.nx])
R | control input weighting matrix (positive-definite matrix of dimension; [obj.nu,obj.nu])
flag_invariant | (optional), see description - if omitted or false, the user-defined order is chosen as opts.termRegOrder (logical)

## Output Arguments

PIset | approximation of the positively invariant set (class: termRegNonlinSysLinApproach)

## See also

-

## References

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