Modelica.Math.Matrices.Utilities

Utility functions that should not be directly utilized by the user

Information

This package contains utility functions that are utilized by higher level matrix functions. These functions are usually not useful for an end-user.

Extends from Modelica.Icons.UtilitiesPackage (Icon for utility packages).

Package Content

Name	Description
continuousRiccatiIterative	Newton's method with exact line search for iterative solving continuous algebraic Riccati equation
discreteRiccatiIterative	Newton's method with exact line search for solving discrete algebraic Riccati equation
toUpperHessenberg	Transform a real square matrix A to upper Hessenberg form H by orthogonal similarity transformation: Q' * A * Q = H
eigenvaluesHessenberg	Compute eigenvalues of an upper Hessenberg form matrix
reorderRSF	Reorders a real Schur form to clusters of stable and unstable eigenvalues
findLocal_tk	Find a local minimizer tk to define the length of the step tk*Nk in continuousRiccatiIterative and discreteRiccatiIterative

Modelica.Math.Matrices.Utilities.continuousRiccatiIterative

Newton's method with exact line search for iterative solving continuous algebraic Riccati equation

Information

Syntax

     X = Matrices.Utilities.continuousRiccatiIterative(A, B, R, Q, X0);
(X, r) = Matrices.Utilities.continuousRiccatiIterative(A, B, R, Q, X0, maxSteps, eps);

Description

This function provides a Newton-like method for solving continuous algebraic Riccati equations (care). It utilizes Exact Line Search to improve the sometimes erratic convergence of Newton's method. Exact line search in this case means, that at each iteration i a Newton step delta_i

X_i+1 = X_i + delta_i

is taken in the direction to minimize the Frobenius norm of the residual

r = || X_i+1*A +A'*X_i+1 - X_i+1*G*X_i+1 + Q ||.

with

      -1
G = B*R *B'

The inputs "maxSteps" and "eps" specify the termination of the iteration. The iteration is terminated if either maxSteps iteration steps have been performed or the relative change delta_i/X_i became smaller than eps.

With an appropriate initial value X0 a sufficiently accurate solution might be reach within a few iteration steps. Although a Lyapunov equation of order n (n is the order of the Riccati equation) is to be solved at each iteration step, the algorithm might be faster than a direct method like Matrices.continuousRiccati, since direct methods have to solve the 2*n-order Hamiltonian system equation.
The algorithm is taken from [1] and [2].

References

[1] Benner, P., Byers, R.
    An Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations
    IEEE Transactions On Automatic Control, Vol. 43, No. 1, pp. 101-107, 1998.
[2] Datta, B.N.
    Numerical Methods for Linear Control Systems
    Elsevier Academic Press, 2004.

Example

A=[0.0,         1.0,         0.0,         0.0;
   0.0,        -1.890,       3.900e-01,  -5.530;
   0.0,        -3.400e-02,  -2.980,       2.430;
   3.400e-02,  -1.100e-03,  -9.900e-01,  -2.100e-01];

B=[ 0.0,         0.0;
    3.600e-01,  -1.60;
   -9.500e-01,  -3.200e-02;
    3.000e-02,   0.0];

R=[1, 0; 0, 1];

Q=[2.313,       2.727,       6.880e-01,   2.300e-02;
   2.727,       4.271,       1.148,       3.230e-01;
   6.880e-01,   1.148,       3.130e-01,   1.020e-01;
   2.300e-02,   3.230e-01,   1.020e-01,   8.300e-02];

X0=identity(4);

(X,r) = Matrices.Utilities.continuousRiccatiIterative(A, B, R, Q, X0);

// X = [1.3239,  0.9015,  0.5466, -1.7672;
        0.9015,  0.9607,  0.4334, -1.1989;
        0.5466,  0.4334,  0.4605, -1.3633;
       -1.7672, -1.1989, -1.3633,  4.4612]
// r =  2.48809423389491E-015

(,r) = Matrices.Utilities.continuousRiccatiIterative(A, B, R, Q, X0,4);

// r =  0.0004;

See also

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Matrices.Utilities.discreteRiccatiIterative

Newton's method with exact line search for solving discrete algebraic Riccati equation

Information

Syntax

     X = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0);
(X, r) = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0, maxSteps, eps);

Description

This function provides a Newton-like method for solving discrete-time algebraic Riccati equations. It uses Exact Line Search to improve the sometimes erratic convergence of Newton's method. Exact line search in this case means, that at each iteration i a Newton step delta_i

X_i+1 = X_i + delta_i

is taken in the direction to minimize the Frobenius norm of the residual

r = || A'X_i+1*A - X_i+1 - A'X_i+1*G_i*X_i+1*A + Q ||

with

                     -1
G_i = B*(R + B'*X_i*B) *B'

Output r is the norm of the residual of the last iteration.

The inputs "maxSteps" and "eps" specify the termination of the iteration. The iteration is terminated if either maxSteps iteration steps have been performed or the relative change delta_i/X_i became smaller than eps.

With an appropriate initial value X0 a sufficiently accurate solution might be reach with a few iteration steps. Although a Lyapunov equation of order n (n is the order of the Riccati equation) is to be solved at each iteration step, the algorithm might be faster than a direct method like Matrices.discreteRiccati, since direct methods have to solve the 2*n-order Hamiltonian system equation. The algorithm is taken from [1] and [2].

References

[1] Benner, P., Byers, R.
    An Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations
    IEEE Transactions On Automatic Control, Vol. 43, No. 1, pp. 101-107, 1998.
[2] Datta, B.N.
    Numerical Methods for Linear Control Systems
    Elsevier Academic Press, 2004.

Example

A  = [0.9970,    0.0000,    0.0000,    0.0000;
      1.0000,    0.0000,    0.0000,    0.0000;
      0.0000,    1.0000,    0.0000,    0.0000;
      0.0000,    0.0000,    1.0000,    0.0000];

B  = [0.0150;
      0.0000;
      0.0000;
      0.0000];

R = [0.2500];

Q = [0, 0, 0, 0;
     0, 0, 0, 0;
     0, 0, 0, 0;
     0, 0, 0, 1];

X0=identity(4);

(X,r) = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0);

//  X = [30.625, 0.0, 0.0, 0.0;
          0.0,   1.0, 0.0, 0.0;
          0.0,   0.0, 1.0, 0.0;
          0.0,   0.0, 0.0, 1.0];

// r =   3.10862446895044E-015

See also

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Matrices.Utilities.toUpperHessenberg

Transform a real square matrix A to upper Hessenberg form H by orthogonal similarity transformation: Q' * A * Q = H

Information

Syntax

                H = Matrices.Utilities.toUpperHessenberg(A);
(H, V, tau, info) = Matrices.Utilities.toUpperHessenberg(A,ilo, ihi);

Description

Function toUpperHessenberg computes a upper Hessenberg form H of a matrix A by orthogonal similarity transformation: Q' * A * Q = H. The optional inputs ilo and ihi improve efficiency if the matrix is already partially converted to Hessenberg form; it is assumed that matrix A is already upper Hessenberg for rows and columns 1:(ilo-1) and (ihi+1):size(A, 1). The function calls LAPACK.dgehrd. See Matrices.LAPACK.dgehrd for more information about the additional outputs V, tau, info and inputs ilo, ihi.

Example

A  = [1, 2, 3;
      6, 5, 4;
      1, 0, 0];

H = toUpperHessenberg(A);

results in:

H = [1.0,  -2.466,  2.630;
    -6.083, 5.514, -3.081;
     0.0,   0.919, -0.514]

See also

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Matrices.Utilities.eigenvaluesHessenberg

Compute eigenvalues of an upper Hessenberg form matrix

Information

Syntax

        ev = Matrices.Utilities.eigenvaluesHessenberg(H);
(ev, info) = Matrices.Utilities.eigenvaluesHessenberg(H);

Description

This function computes the eigenvalues of a Hessenberg form matrix. Transformation to Hessenberg form is the first step in eigenvalue computation for arbitrary matrices with QR decomposition. This step can be skipped if the matrix has already Hessenberg form.

The function uses the LAPACK-routine dhseqr. Output info is 0 for a successful call of this function.
See Matrices.LAPACK.dhseqr for details

Example

Real A[3,3] = [1,2,3;
               9,8,7;
               0,1,0];

Real ev[3,2];

ev := Matrices.Utilities.eigenvaluesHessenberg(A);

// ev  = [10.7538,    0.0;
          -0.8769,    1.0444;
          -0.8769,   -1.0444]
// = {10.7538,  -0.8769 +- i*1.0444}

See also

Matrices.eigenValues, Matrices.hessenberg

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Matrices.Utilities.reorderRSF

Reorders a real Schur form to clusters of stable and unstable eigenvalues

Information

Syntax

              To = Matrices.Utilities.reorderRSF(T, Q, alphaReal, alphaImag);
(To, Qo, wr, wi) = Matrices.Utilities.reorderRSF(T, Q, alphaReal, alphaImag, iscontinuous);

Description

Function reorderRSF() reorders a real Schur form such that the stable eigenvalues of the system are in the 1-by-1 and 2-by-2 diagonal blocks of the block upper triangular matrix. If the Schur form is referenced to a continuous system the staple eigenvalues are in the left complex half plane. The stable eigenvalues of a discrete system are inside the complex unit circle.
This function is used for example to solve algebraic Riccati equations (continuousRiccati, discreteRiccati). In this context the Schur form as well as the corresponding eigenvalues and the transformation matrix Q are known, why the eigenvalues and the transformation matrix are inputs to reorderRSF().
The Schur vector matrix Qo is also reordered according to To. The vectors wr and wi contains the real and imaginary parts of the reordered eigenvalues respectively.

Example

T := [-1,2, 3,4;
       0,2, 6,5;
       0,0,-3,5;
       0,0, 0,6];
To := Matrices.Utilities.reorderRSF(T,identity(4),{-1, 2, -3, 6},{0, 0, 0, 0}, true);

// To = [-1.0, -0.384, 3.585, 4.0;
//        0.0, -3.0,   6.0,   0.64;
//        0.0,  0.0,   2.0,   7.04;
//        0.0,  0.0,   0.0,   6.0]

See also Matrices.realSchur

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Matrices.Utilities.findLocal_tk

Find a local minimizer tk to define the length of the step tk*Nk in continuousRiccatiIterative and discreteRiccatiIterative

Information

Syntax

tk = Matrices.Utilities.findLocal_tk(Rk, Vk);

Description

Function findLocal_tk() is an auxiliary function called in iterative solver for algebraic Riccati equation based on Newton's method with exact line search like continuousRiccatiIterative
and discreteRiccatiIterative.
The function computes the local minimum of the function f_k(t_k)

f_k(t_k) = alpha_k*(1-t_k)^2 + 2*beta_k*(1-t)*t^2 + gamma_k*t^4

by calculating the zeros of the derivation d f_k/d t_k. It is known that the function f_k(t_k) has a local minimum at some value t_k_min in [0, 2].
With t_k_min the norm of the next residual of the algorithm will be minimized.
See [1] for more information

References

[1] Benner, P., Byers, R.
    An Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations
    IEEE Transactions On Automatic Control, Vol. 43, No. 1, pp. 101-107, 1998.

See also

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs