%% initialGuessSet_RiccatiEq
% solves the Riccati equation for the linearized
% dynamics and encloses the resulting ellipsoid by a zonotope.
%
%% Syntax
%
% <html>
%        <div class="syntax">
% PIset = initialGuessSet_RiccatiEq(obj,dynamics,Q,R) <br>
% PIset = initialGuessSet_RiccatiEq(obj,dynamics,Q,R,flag_invariant) <br>
%        </div>
% </html>
%
%% 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
%
% <html>
% <table cellspacing="0" cellpadding="4" width="" border="1" frame="box" rules="none" class="">
% <tbody valign="top">
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            obj
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            instance of class
%            computeTermRegNonlinSysOptApproach
%        </p>
%    </td>
%    </tr>
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            dynamics
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            instance of class termRegNonlinSysDynamics
%        </p>
%    </td>
%    </tr>
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            Q
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            state weighting matrix
%            (positive-definite matrix of dimension;
%            [obj.nx,obj.nx])
%        </p>
%    </td>
%    </tr>
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            R
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            control input weighting matrix
%            (positive-definite matrix of dimension;
%            [obj.nu,obj.nu])
%        </p>
%    </td>
%    </tr>
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            flag_invariant
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            (optional), see description - if omitted or
%            false, the user-defined order is chosen as
%            opts.termRegOrder
%            (logical)
%        </p>
%    </td>
%    </tr>
% </tbody>
% </table>
% </html>
%
%% Output Arguments
%
% <html>
% <table cellspacing="0" cellpadding="4" width="" border="1" frame="box" rules="none" class="">
% <tbody valign="top">
%    <tr bgcolor="#F2F2F2">
%    <td>
%        <p class="table">
%            PIset
%        </p>
%    </td>
%    <td>
%        <p class="table">
%            approximation of the positively invariant set
%            (class: termRegNonlinSysLinApproach)
%        </p>
%    </td>
%    </tr>
% </tbody>
% </table>
% </html>
%
%% See also
% <matlab:doc('-') ->
%
%% References
% -
%

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