Modelica.Math.Matrices

Library of functions operating on matrices

Information

Library content

This library provides functions operating on matrices. Below, the functions are ordered according to categories and a typical call of the respective function is shown. Most functions are solely an interface to the external LAPACK library.

Note: A' is a short hand notation of transpose(A):

Basic Information

• toString(A) - returns the string representation of matrix A.
• isEqual(M1, M2) - returns true if matrices M1 and M2 have the same size and the same elements.

Linear Equations

• solve(A,b) - returns solution x of the linear equation A*x=b (where b is a vector, and A is a square matrix that must be regular).
• solve2(A,B) - returns solution X of the linear equation A*X=B (where B is a matrix, and A is a square matrix that must be regular)
• leastSquares(A,b) - returns solution x of the linear equation A*x=b in a least squares sense (where b is a vector and A may be non-square and may be rank deficient)
• leastSquares2(A,B) - returns solution X of the linear equation A*X=B in a least squares sense (where B is a matrix and A may be non-square and may be rank deficient)
• equalityLeastSquares(A,a,B,b) - returns solution x of a linear equality constrained least squares problem: min|A*x-a|^2 subject to B*x=b
• (LU,p,info) = LU(A) - returns the LU decomposition with row pivoting of a rectangular matrix A.
• LU_solve(LU,p,b) - returns solution x of the linear equation L*U*x[p]=b with a b vector and an LU decomposition from "LU(..)".
• LU_solve2(LU,p,B) - returns solution X of the linear equation L*U*X[p,:]=B with a B matrix and an LU decomposition from "LU(..)".

Matrix Factorizations

• (eval,evec) = eigenValues(A) - returns eigen values "eval" and eigen vectors "evec" for a real, nonsymmetric matrix A in a Real representation.
• eigenValueMatrix(eval) - returns real valued block diagonal matrix of the eigenvalues "eval" of matrix A.
• (sigma,U,VT) = singularValues(A) - returns singular values "sigma" and left and right singular vectors U and VT of a rectangular matrix A.
• (Q,R,p) = QR(A) - returns the QR decomposition with column pivoting of a rectangular matrix A such that Q*R = A[:,p].
• (H,U) = hessenberg(A) - returns the upper Hessenberg form H and the orthogonal transformation matrix U of a square matrix A such that H = U'*A*U.
• realSchur(A) - returns the real Schur form of a square matrix A.
• cholesky(A) - returns the cholesky factor H of a real symmetric positive definite matrix A so that A = H'*H.
• (D,Aimproved) = balance(A) - returns an improved form Aimproved of a square matrix A that has a smaller condition as A, with Aimproved = inv(diagonal(D))*A*diagonal(D).

Matrix Properties

• trace(A) - returns the trace of square matrix A, i.e., the sum of the diagonal elements.
• det(A) - returns the determinant of square matrix A (using LU decomposition; try to avoid det(..))
• inv(A) - returns the inverse of square matrix A (try to avoid, use instead "solve2(..) with B=identity(..))
• rank(A) - returns the rank of square matrix A (computed with singular value decomposition)
• conditionNumber(A) - returns the condition number norm(A)*norm(inv(A)) of a square matrix A in the range 1..∞.
• rcond(A) - returns the reciprocal condition number 1/conditionNumber(A) of a square matrix A in the range 0..1.
• norm(A) - returns the 1-, 2-, or infinity-norm of matrix A.
• frobeniusNorm(A) - returns the Frobenius norm of matrix A.
• nullSpace(A) - returns the null space of matrix A.

Matrix Exponentials

• exp(A) - returns the exponential e^A of a matrix A by adaptive Taylor series expansion with scaling and balancing
• (phi, gamma) = integralExp(A,B) - returns the exponential phi=e^A and the integral gamma=integral(exp(A*t)*dt)*B as needed for a discretized system with zero order hold.
• (phi, gamma, gamma1) = integralExpT(A,B) - returns the exponential phi=e^A, the integral gamma=integral(exp(A*t)*dt)*B, and the time-weighted integral gamma1 = integral((T-t)*exp(A*t)*dt)*B as needed for a discretized system with first order hold.

Matrix Equations

• continuousLyapunov(A,C) - returns solution X of the continuous-time Lyapunov equation X*A + A'*X = C
• continuousSylvester(A,B,C) - returns solution X of the continuous-time Sylvester equation A*X + X*B = C
• continuousRiccati(A,B,R,Q) - returns solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0
• discreteLyapunov(A,C) - returns solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C
• discreteSylvester(A,B,C) - returns solution X of the discrete-time Sylvester equation A*X*B + sgn*X = C
• discreteRiccati(A,B,R,Q) - returns solution X of the discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

Matrix Manipulation

• sort(M) - returns the sorted rows or columns of matrix M in ascending or descending order.
• flipLeftRight(M) - returns matrix M so that the columns of M are flipped in left/right direction.
• flipUpDown(M) - returns matrix M so that the rows of M are flipped in up/down direction.

See also

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
Examples	Examples demonstrating the usage of the Math.Matrices functions
toString	Convert a matrix into its string representation
isEqual	Compare whether two Real matrices are identical
solve	Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)
solve2	Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)
leastSquares	Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
leastSquares2	Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
equalityLeastSquares	Solve a linear equality constrained least squares problem
LU	LU decomposition of square or rectangular matrix
LU_solve	Solve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..))
LU_solve2	Solve real system of linear equations P*L*U*X=B with a B matrix and an LU decomposition (from LU(..))
eigenValues	Return eigenvalues and eigenvectors for a real, nonsymmetric matrix in a Real representation
eigenValueMatrix	Return real valued block diagonal matrix J of eigenvalues of matrix A (A=V*J*Vinv)
singularValues	Return singular values and left and right singular vectors
QR	Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)
hessenberg	Return upper Hessenberg form of a matrix
realSchur	Return the real Schur form (rsf) S of a square matrix A, A=QZ*S*QZ'
cholesky	Return the Cholesky factorization of a symmetric positive definite matrix
balance	Return a balanced form of matrix A to improve the condition of A
balanceABC	Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation
trace	Return the trace of matrix A, i.e., the sum of the diagonal elements
det	Return determinant of a matrix (computed by LU decomposition; try to avoid det(..))
inv	Return inverse of a matrix (try to avoid inv(..))
rank	Return rank of a rectangular matrix (computed with singular values)
conditionNumber	Return the condition number norm(A)*norm(inv(A)) of a matrix A
rcond	Return the reciprocal condition number of a matrix
norm	Return the p-norm of a matrix
frobeniusNorm	Return the Frobenius norm of a matrix
nullSpace	Return the orthonormal nullspace of a matrix
exp	Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing
integralExp	Return the exponential and the integral of the exponential of a matrix
integralExpT	Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix
continuousLyapunov	Return solution X of the continuous-time Lyapunov equation X*A + A'*X = C
continuousSylvester	Return solution X of the continuous-time Sylvester equation A*X + X*B = C
continuousRiccati	Return solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0 (care)
discreteLyapunov	Return solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C
discreteSylvester	Return solution of the discrete-time Sylvester equation A*X*B + sgn*X = C
discreteRiccati	Return solution of discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0 (dare)
sort	Sort the rows or columns of a matrix in ascending or descending order
flipLeftRight	Flip the columns of a matrix in left/right direction
flipUpDown	Flip the rows of a matrix in up/down direction
LAPACK	Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)
Utilities	Utility functions that should not be directly utilized by the user

Modelica.Math.Matrices.toString

Convert a matrix into its string representation

Information

Syntax

Matrices.toString(A);
Matrices.toString(A, name="", significantDigits=6);

Description

The function call "Matrices.toString(A)" returns the string representation of matrix A. With the optional arguments "name" and "significantDigits", a name and the number of the digits are defined. The default values of name and significantDigits are "" and 6 respectively. If name=="" then the prefix "<name> =" is left out.

Example

A = [2.12, -4.34; -2.56, -1.67];

toString(A);
// = "
//      2.12   -4.34
//     -2.56   -1.67";

toString(A,"A",1);
// = "A =
//         2     -4
//        -3     -2"

See also

Vectors.toString

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

Inputs

Outputs

Modelica.Math.Matrices.isEqual

Compare whether two Real matrices are identical

Information

Syntax

Matrices.isEqual(M1, M2);
Matrices.isEqual(M1, M2, eps=0);

Description

The function call "Matrices.isEqual(M1, M2)" returns true, if the two Real matrices M1 and M2 have the same dimensions and the same elements. Otherwise the function returns false. Two elements e1 and e2 of the two matrices are checked on equality by the test "abs(e1-e2) ≤ eps", where "eps" can be provided as third argument of the function. Default is "eps = 0".

Example

  Real A1[2,2] = [1,2; 3,4];
  Real A2[3,2] = [1,2; 3,4; 5,6];
  Real A3[2,2] = [1,2, 3,4.0001];
  Boolean result;
algorithm
  result := Matrices.isEqual(M1,M2);     // = false
  result := Matrices.isEqual(M1,M3);     // = false
  result := Matrices.isEqual(M1,M1);     // = true
  result := Matrices.isEqual(M1,M3,0.1); // = true

See also

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

Inputs

Outputs

Modelica.Math.Matrices.solve

Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)

Information

Syntax

Matrices.solve(A,b);

Description

This function call returns the solution x of the linear system of equations

A*x = b

If a unique solution x does not exist (since A is singular), an assertion is triggered. If this is not desired, use instead Matrices.leastSquares and inquire the singularity of the solution with the return argument rank (a unique solution is computed if rank = size(A,1)).

Note, the solution is computed with the LAPACK function "dgesv", i.e., by Gaussian elimination with partial pivoting.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b[3] = {10,22,12};
  Real x[3];
algorithm
  x := Matrices.solve(A,b);  // x = {3,2,1}

See also

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

Inputs

Outputs

Modelica.Math.Matrices.solve2

Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)

Information

Syntax

Matrices.solve2(A,b);

Description

This function call returns the solution X of the linear system of equations

A*X = B

If a unique solution X does not exist (since A is singular), an assertion is triggered. If this is not desired, use instead Matrices.leastSquares2 and inquire the singularity of the solution with the return argument rank (a unique solution is computed if rank = size(A,1)).

Note, the solution is computed with the LAPACK function "dgesv", i.e., by Gaussian elimination with partial pivoting.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real B[3,2] = [10, 20;
                 22, 44;
                 12, 24];
  Real X[3,2];
algorithm
  X := Matrices.solve2(A, B);  /* X = [3, 6;
                                       2, 4;
                                       1, 2] */

See also

Matrices.LU, Matrices.LU_solve2, Matrices.leastSquares2.

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

Inputs

Outputs

Modelica.Math.Matrices.leastSquares

Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Information

Syntax

x = Matrices.leastSquares(A,b);

Description

Returns a solution of equation A*x = b in a least square sense (A may be rank deficient):

minimize | A*x - b |

Several different cases can be distinguished (note, rank is an output argument of this function):

size(A,1) = size(A,2)

A solution is returned for a regular, as well as a singular matrix A:

• rank = size(A,1):
  A is regular and the returned solution x fulfills the equation A*x = b uniquely.
• rank < size(A,1):
  A is singular and no unique solution for equation A*x = b exists.
  • If an infinite number of solutions exists, the one is selected that fulfills the equation and at the same time has the minimum norm |x| for all solution vectors that fulfill the equation.
  • If no solution exists, x is selected such that |A*x - b| is as small as possible (but A*x - b is not zero).

size(A,1) > size(A,2):

The equation A*x = b has no unique solution. The solution x is selected such that |A*x - b| is as small as possible. If rank = size(A,2), this minimum norm solution is unique. If rank < size(A,2), there are an infinite number of solutions leading to the same minimum value of |A*x - b|. From these infinite number of solutions, the one with the minimum norm |x| is selected. This gives a unique solution that minimizes both |A*x - b| and |x|.

size(A,1) < size(A,2):

• rank = size(A,1):
  There are an infinite number of solutions that fulfill the equation A*x = b. From this infinite number, the unique solution is selected that minimizes |x|.
• rank < size(A,1):
  There is either no solution of equation A*x = b, or there are again an infinite number of solutions. The unique solution x is returned that minimizes both |A*x - b| and |x|.

Note, the solution is computed with the LAPACK function "dgelsy", i.e., QR or LQ factorization of A with column pivoting.

Algorithmic details

The function first computes a QR factorization with column pivoting:

A * P = Q * [ R11 R12 ]
            [  0  R22 ]

with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:

A * P = Q * [ T11 0 ] * Z
            [  0  0 ]

The minimum-norm solution is then

x = P * Z' [ inv(T11)*Q1'*b ]
           [        0       ]

where Q1 consists of the first "rank" columns of Q.

See also

Matrices.leastSquares2 (same as leastSquares, but with a right hand side matrix),
Matrices.solve (for square, regular matrices A)

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

Inputs

Outputs

Modelica.Math.Matrices.leastSquares2

Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Information

Syntax

X = Matrices.leastSquares2(A,B);

Description

Returns a solution of equation A*X = B in a least square sense (A may be rank deficient):

minimize | A*X - B |

Several different cases can be distinguished (note, rank is an output argument of this function):

size(A,1) = size(A,2)

A solution is returned for a regular, as well as a singular matrix A:

• rank = size(A,1):
  A is regular and the returned solution X fulfills the equation A*X = B uniquely.
• rank < size(A,1):
  A is singular and no unique solution for equation A*X = B exists.
  • If an infinite number of solutions exists, the one is selected that fulfills the equation and at the same time has the minimum norm |x| for all solution vectors that fulfill the equation.
  • If no solution exists, X is selected such that |A*X - B| is as small as possible (but A*X - B is not zero).

size(A,1) > size(A,2):

The equation A*X = B has no unique solution. The solution X is selected such that |A*X - B| is as small as possible. If rank = size(A,2), this minimum norm solution is unique. If rank < size(A,2), there are an infinite number of solutions leading to the same minimum value of |A*X - B|. From these infinite number of solutions, the one with the minimum norm |X| is selected. This gives a unique solution that minimizes both |A*X - B| and |X|.

size(A,1) < size(A,2):

• rank = size(A,1):
  There are an infinite number of solutions that fulfill the equation A*X = B. From this infinite number, the unique solution is selected that minimizes |X|.
• rank < size(A,1):
  There is either no solution of equation A*X = B, or there are again an infinite number of solutions. The unique solution X is returned that minimizes both |A*X - B| and |X|.

Note, the solution is computed with the LAPACK function "dgelsy", i.e., QR or LQ factorization of A with column pivoting.

Algorithmic details

The function first computes a QR factorization with column pivoting:

A * P = Q * [ R11 R12 ]
            [  0  R22 ]

with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:

A * P = Q * [ T11 0 ] * Z
            [  0  0 ]

The minimum-norm solution is then

X = P * Z' [ inv(T11)*Q1'*B ]
           [        0       ]

where Q1 consists of the first "rank" columns of Q.

See also

Matrices.leastSquares (same as leastSquares2, but with a right hand side vector),
Matrices.solve2 (for square, regular matrices A)

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

Inputs

Outputs

Modelica.Math.Matrices.equalityLeastSquares

Solve a linear equality constrained least squares problem

Information

Syntax

x = Matrices.equalityLeastSquares(A,a,B,b);

Description

This function returns the solution x of the linear equality-constrained least squares problem:

min|A*x - a|^2 over x, subject to B*x = b

It is required that the dimensions of A and B fulfill the following relationship:

size(B,1) ≤ size(A,2) ≤ size(A,1) + size(B,1)

Note, the solution is computed with the LAPACK function "dgglse" using the generalized RQ factorization under the assumptions that B has full row rank (= size(B,1)) and the matrix [A;B] has full column rank (= size(A,2)). In this case, the problem has a unique solution.

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

Inputs

Outputs

Modelica.Math.Matrices.LU

LU decomposition of square or rectangular matrix

Information

Syntax

(LU, pivots)       = Matrices.LU(A);
(LU, pivots, info) = Matrices.LU(A);

Description

This function call returns the LU decomposition of a "Real[m,n]" matrix A, i.e.,

P*L*U = A

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). Matrices L and U are stored in the returned matrix LU (the diagonal of L is not stored). With the companion function Matrices.LU_solve, this decomposition can be used to solve linear systems (P*L*U)*x = b with different right hand side vectors b. If a linear system of equations with just one right hand side vector b shall be solved, it is more convenient to just use the function Matrices.solve.

The optional third (Integer) output argument has the following meaning:

The LU factorization is computed with the LAPACK function "dgetrf", i.e., by Gaussian elimination using partial pivoting with row interchanges. Vector "pivots" are the pivot indices, i.e., for 1 ≤ i ≤ min(m,n), row i of matrix A was interchanged with row pivots[i].

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b1[3] = {10,22,12};
  Real b2[3] = { 7,13,10};
  Real    LU[3,3];
  Integer pivots[3];
  Real    x1[3];
  Real    x2[3];
algorithm
  (LU, pivots) := Matrices.LU(A);
  x1 := Matrices.LU_solve(LU, pivots, b1);  // x1 = {3,2,1}
  x2 := Matrices.LU_solve(LU, pivots, b2);  // x2 = {1,0,2}

See also

Matrices.LU_solve, Matrices.solve,

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

Inputs

Outputs

Modelica.Math.Matrices.LU_solve

Solve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..))

Information

Syntax

Matrices.LU_solve(LU, pivots, b);

Description

This function call returns the solution x of the linear systems of equations

P*L*U*x = b;

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). The matrices of this decomposition are computed with function Matrices.LU that returns arguments LU and pivots used as input arguments of Matrices.LU_solve. With Matrices.LU and Matrices.LU_solve it is possible to efficiently solve linear systems with different right hand side vectors. If a linear system of equations with just one right hand side vector shall be solved, it is more convenient to just use the function Matrices.solve.

If a unique solution x does not exist (since the LU decomposition is singular), an exception is raised.

The LU factorization is computed with the LAPACK function "dgetrf", i.e., by Gaussian elimination using partial pivoting with row interchanges. Vector "pivots" are the pivot indices, i.e., for 1 ≤ i ≤ min(m,n), row i of matrix A was interchanged with row pivots[i].

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b1[3] = {10,22,12};
  Real b2[3] = { 7,13,10};
  Real    LU[3,3];
  Integer pivots[3];
  Real    x1[3];
  Real    x2[3];
algorithm
  (LU, pivots) := Matrices.LU(A);
  x1 := Matrices.LU_solve(LU, pivots, b1);  // x1 = {3,2,1}
  x2 := Matrices.LU_solve(LU, pivots, b2);  // x2 = {1,0,2}

See also

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

Inputs

Outputs

Modelica.Math.Matrices.LU_solve2

Solve real system of linear equations P*L*U*X=B with a B matrix and an LU decomposition (from LU(..))

Information

Syntax

Matrices.LU_solve2(LU, pivots, B);

Description

This function call returns the solution X of the linear systems of equations

P*L*U*X = B;

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). The matrices of this decomposition are computed with function Matrices.LU that returns arguments LU and pivots used as input arguments of Matrices.LU_solve2. With Matrices.LU and Matrices.LU_solve2 it is possible to efficiently solve linear systems with different right hand side matrices. If a linear system of equations with just one right hand side matrix shall be solved, it is more convenient to just use the function Matrices.solve2.

If a unique solution X does not exist (since the LU decomposition is singular), an exception is raised.

The LU factorization is computed with the LAPACK function "dgetrf", i.e., by Gaussian elimination using partial pivoting with row interchanges. Vector "pivots" are the pivot indices, i.e., for 1 ≤ i ≤ min(m,n), row i of matrix A was interchanged with row pivots[i].

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real B1[3] = [10, 20;
                22, 44;
                12, 24];
  Real B2[3] = [ 7, 14;
                13, 26;
                10, 20];
  Real    LU[3,3];
  Integer pivots[3];
  Real    X1[3,2];
  Real    X2[3,2];
algorithm
  (LU, pivots) := Matrices.LU(A);
  X1 := Matrices.LU_solve2(LU, pivots, B1);  /* X1 = [3, 6;
                                                      2, 4;
                                                      1, 2] */
  X2 := Matrices.LU_solve2(LU, pivots, B2);  /* X2 = [1, 2;
                                                      0, 0;
                                                      2, 4] */

See also

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

Inputs

Outputs

Modelica.Math.Matrices.eigenValues

Return eigenvalues and eigenvectors for a real, nonsymmetric matrix in a Real representation

Information

Syntax

                eigenvalues = Matrices.eigenValues(A);
(eigenvalues, eigenvectors) = Matrices.eigenValues(A);

Description

This function call returns the eigenvalues and optionally the (right) eigenvectors of a square matrix A. The first column of "eigenvalues" contains the real and the second column contains the imaginary part of the eigenvalues. If the i-th eigenvalue has no imaginary part, then eigenvectors[:,i] is the corresponding real eigenvector. If the i-th eigenvalue has an imaginary part, then eigenvalues[i+1,:] is the conjugate complex eigenvalue and eigenvectors[:,i] is the real and eigenvectors[:,i+1] is the imaginary part of the eigenvector of the i-th eigenvalue. With function Matrices.eigenValueMatrix, a real block diagonal matrix is constructed from the eigenvalues such that

A = eigenvectors * eigenValueMatrix(eigenvalues) * inv(eigenvectors)

provided the eigenvector matrix "eigenvectors" can be inverted (an inversion is possible, if all eigenvalues are different; in some cases, an inversion is also possible if some eigenvalues are the same).

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real eval[3,2];
algorithm
  eval := Matrices.eigenValues(A);  // eval = [-0.618, 0;
                                    //          8.0  , 0;
                                    //          1.618, 0];

i.e., matrix A has the 3 real eigenvalues -0.618, 8, 1.618.

See also

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

Inputs

Outputs

Modelica.Math.Matrices.eigenValueMatrix

Return real valued block diagonal matrix J of eigenvalues of matrix A (A=V*J*Vinv)

Information

Syntax

Matrices.eigenValueMatrix(eigenvalues);

Description

The function call returns a block diagonal matrix J from the two-column matrix eigenvalues (computed by function Matrices.eigenValues). Matrix eigenvalues must have the real part of the eigenvalues in the first column and the imaginary part in the second column. If an eigenvalue i has a vanishing imaginary part, then J[i,i] = eigenvalues[i,1], i.e., the diagonal element of J is the real eigenvalue. Otherwise, eigenvalue i and conjugate complex eigenvalue i+1 are used to construct a 2 by 2 diagonal block of J:

J[i  , i]   := eigenvalues[i,1];
J[i  , i+1] := eigenvalues[i,2];
J[i+1, i]   := eigenvalues[i+1,2];
J[i+1, i+1] := eigenvalues[i+1,1];

See also

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

Inputs

Outputs

Modelica.Math.Matrices.singularValues

Return singular values and left and right singular vectors

Information

Syntax

         sigma = Matrices.singularValues(A);
(sigma, U, VT) = Matrices.singularValues(A);

Description

This function computes the singular values and optionally the singular vectors of matrix A. Basically the singular value decomposition of A is computed, i.e.,

A = U Σ VT
  = U*Sigma*VT

where U and V are orthogonal matrices (UU^T=I, VV^T=I). Σ = [diagonal(σ_i), zeros(n,m-n)], if n=size(A,1) ≤ m=size(A,2)) or [diagonal(σ_i); zeros(n-m,m)], if n > m=size(A,2)). Σ has the same size as matrix A with nonnegative diagonal elements in decreasing order and with all other elements zero (σ₁ is the largest element). The function returns the singular values σ_i in vector sigma and the orthogonal matrices in matrices U and VT.

Example

A = [1, 2,  3,  4;
     3, 4,  5, -2;
    -1, 2, -3,  5];
(sigma, U, VT) = singularValues(A);
results in:
   sigma = {8.33, 6.94, 2.31};
i.e.
   Sigma = [8.33,    0,    0, 0;
               0, 6.94,    0, 0;
               0,    0, 2.31, 0]

See also

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

Inputs

Outputs

Modelica.Math.Matrices.QR

Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)

Information

Syntax

(Q,R,p) = Matrices.QR(A);

Description

This function returns the QR decomposition of a rectangular matrix A (the number of columns of A must be less than or equal to the number of rows):

Q*R = A[:,p]

where Q is a rectangular matrix that has orthonormal columns and has the same size as A (Q^TQ=I), R is a square, upper triangular matrix and p is a permutation vector. Matrix R has the following important properties:

• The absolute value of a diagonal element of R is the largest value in this row, i.e., abs(R[i,i]) ≥ abs(R[i,j]).
• The diagonal elements of R are sorted according to size, such that the largest absolute value is abs(R[1,1]) and abs(R[i,i]) ≥ abs(R[j,j]) with i < j.

This means that if abs(R[i,i]) ≤ ε then abs(R[j,k]) ≤ ε for j ≥ i, i.e., the i-th row up to the last row of R have small elements and can be treated as being zero. This allows to, e.g., estimate the row-rank of R (which is the same row-rank as A). Furthermore, R can be partitioned in two parts

A[:,p] = Q * [R1, R2;
              0,  0]

where R₁ is a regular, upper triangular matrix.

Note, the solution is computed with the LAPACK functions "dgeqp3" and "dorgqr", i.e., by Householder transformations with column pivoting. If Q is not needed, the function may be called as: (,R,p) = QR(A).

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real R[3,3];
algorithm
  (,R) := Matrices.QR(A);  // R = [-7.07.., -4.24.., -3.67..;
                                    0     , -1.73.., -0.23..;
                                    0     ,  0     ,  0.65..];

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

Inputs

Outputs

Modelica.Math.Matrices.hessenberg

Return upper Hessenberg form of a matrix

Information

Syntax

     H = Matrices.hessenberg(A);
(H, U) = Matrices.hessenberg(A);
 

Description

Function hessenberg computes the Hessenberg matrix H of matrix A as well as the orthogonal transformation matrix U that holds H = U'*A*U. The Hessenberg form of a matrix is computed by repeated Householder similarity transformation. The elementary reflectors and the corresponding scalar factors are provided by function "Utilities.toUpperHessenberg()". The transformation matrix U is then computed by LAPACK.dorghr.

Example

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

(H, U) = hessenberg(A);

results in:

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

U = [1.0,    0.0,      0.0;
     0.0,   -0.9864,  -0.1644;
     0.0,   -0.1644,   0.9864]

and therefore,

U*H*transpose(U) = [1.0, 2.0, 3.0;
                    6.0, 5.0, 4.0;
                    1.0, 0.0, 0.0]

See also

Matrices.Utilities.toUpperHessenberg

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

Inputs

Outputs

Modelica.Math.Matrices.realSchur

Return the real Schur form (rsf) S of a square matrix A, A=QZ*S*QZ'

Information

Syntax

                            S = Matrices.realSchur(A);
(S, QZ, alphaReal, alphaImag) = Matrices.realSchur(A);

Description

Function realSchur calculates the real Schur form of a real square matrix A, i.e.

A = QZ*S*transpose(QZ)

with the real nxn matrices S and QZ. S is a block upper triangular matrix with 1x1 and 2x2 blocks in the diagonal. QZ is an orthogonal matrix. The 1x1 blocks contains the real eigenvalues of A. The 2x2 blocks [s11, s12; s21, s11] represents the conjugated complex pairs of eigenvalues, whereas the real parts of the eigenvalues are the elements of the diagonal (s11). The imaginary parts are the positive and negative square roots of the product of the two elements s12 and s21 (imag = +-sqrt(s12*s21)).

The calculation in lapack.dgees is performed stepwise, i.e., using the internal methods of balancing and scaling of dgees.

Example

   Real A[3,3] = [1, 2, 3; 4, 5, 6; 7, 8, 9];
   Real T[3,3];
   Real Z[3,3];
   Real alphaReal[3];
   Real alphaImag[3];

algorithm
  (T, Z, alphaReal, alphaImag):=Modelica.Math.Matrices.realSchur(A);
//   T = [16.12, 4.9,   1.59E-015;
//        0,    -1.12, -1.12E-015;
//        0,     0,    -1.30E-015]
//   Z = [-0.23,  -0.88,   0.41;
//        -0.52,  -0.24,  -0.82;
//        -0.82,   0.4,    0.41]
//alphaReal = {16.12, -1.12, -1.32E-015}
//alphaImag = {0, 0, 0}

See also

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

Inputs

Outputs

Modelica.Math.Matrices.cholesky

Return the Cholesky factorization of a symmetric positive definite matrix

Information

Syntax

H = Matrices.cholesky(A);
H = Matrices.cholesky(A, upper=true);

Description

Function cholesky computes the Cholesky factorization of a real symmetric positive definite matrix A. The optional Boolean input "upper" specifies whether the upper or the lower triangular matrix is returned, i.e.

A = H'*H   if upper is true (H is upper triangular)
A = H*H'   if upper is false (H is lower triangular)

The computation is performed by LAPACK.dpotrf.

Example

A  = [1, 0,  0;
      6, 5,  0;
      1, -2,  2];
S = A*transpose(A);

H = Matrices.cholesky(S);

results in:

H = [1.0,  6.0,  1.0;
     0.0,  5.0, -2.0;
     0.0,  0.0,  2.0]

with

transpose(H)*H = [1.0,  6.0,   1;
                  6.0, 61.0,  -4.0;
                  1.0, -4.0,   9.0] //=S

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

Inputs

Outputs

Modelica.Math.Matrices.balance

Return a balanced form of matrix A to improve the condition of A

Information

Syntax

(D,B) = Matrices.balance(A);

Description

This function returns a vector D, such that B=inv(diagonal(D))*A*diagonal(D) has a better condition as matrix A, i.e., conditionNumber(B) ≤ conditionNumber(A). The elements of D are multiples of 2 which means that this function does not introduce round-off errors. Balancing attempts to make the norm of each row of B equal to the norm of the respective column.

Balancing is used to minimize roundoff errors induced through large matrix calculations like Taylor-series approximation or computation of eigenvalues.

Example

- A = [1, 10,  1000; 0.01,  0,  10; 0.005,  0.01,  10]
- Matrices.norm(A, 1);
  = 1020.0
- (T,B)=Matrices.balance(A)
- T
  = {256, 16, 0.5}
- B
  =  [1,     0.625,   1.953125;
      0.16,  0,       0.3125;
      2.56,  0.32,   10.0]
- Matrices.norm(B, 1);
  = 12.265625

The Algorithm is taken from

H. D. Joos, G. Grübel:

RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which based on the balance function from EISPACK.

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

Inputs

Outputs

Modelica.Math.Matrices.balanceABC

Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation

Information

Syntax

(scale,As,Bs,Cs) = Matrices.balanceABC(A,B,C);
(scale,As,Bs)    = Matrices.balanceABC(A,B);
(scale,As,,Cs)   = Matrices.balanceABC(A,C=C);

Description

This function returns a vector scale, such that with T=diagonal(scale) system matrix S_scale

          |inv(T)*A*T, inv(T)*B|
S_scale = |                    |
          |       C*T,     0   |

has a better condition as system matrix S

    |A, B|
S = |    |
    |C, 0|

that is, conditionNumber(S_scale) ≤ conditionNumber(S). The elements of vector scale are multiples of 2 which means that this function does not introduce round-off errors.

Balancing a linear dynamic system in state space form

der(x) = A*x + B*u
    y  = C*x + D*u

means to find a state transformation x_new = T*x = diagonal(scale)*x so that the transformed system is better suited for numerical algorithms.

Example

import Modelica.Math.Matrices;

A = [1, -10,  1000; 0.01,  0,  10; 0.005,  -0.01,  10];
B = [100, 10; 1,0; -0.003, 1];
C = [-0.5, 1, 100];

(scale, As, Bs, Cs) := Matrices.balanceABC(A,B,C);
T    = diagonal(scale);
Diff = [Matrices.inv(T)*A*T, Matrices.inv(T)*B;
        C*T, zeros(1,2)] - [As, Bs; Cs, zeros(1,2)];
err  = Matrices.norm(Diff);

-> Results in:
scale = {16, 1, 0.0625}
norm(A)  = 1000.15, norm(B)  = 100.504, norm(C)  = 100.006
norm(As) = 10.8738, norm(Bs) = 16.0136, norm(Cs) = 10.2011
err = 0

The algorithm is taken from

H. D. Joos, G. Grübel:

RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which is based on the balance function from EISPACK.

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

Inputs

Outputs

Modelica.Math.Matrices.trace

Return the trace of matrix A, i.e., the sum of the diagonal elements

Information

Syntax

r = Matrices.trace(A);

Description

This function computes the trace, i.e., the sum of the elements in the diagonal of matrix A.

Example

A = [1, 3;
     2, 1];
r = trace(A);

results in:

r = 2.0

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

Inputs

Outputs

Modelica.Math.Matrices.det

Return determinant of a matrix (computed by LU decomposition; try to avoid det(..))

Information

Syntax

result = Matrices.det(A);

Description

This function returns the determinant "result" of matrix A computed by a LU decomposition with row pivoting. For details about determinants, see http://en.wikipedia.org/wiki/Determinant. Usually, this function should never be used, because there are nearly always better numerical algorithms as by computing the determinant. Examples:

• Use Matrices.rank to compute whether det(A) = 0 (i.e., Matrices.rank(A) < size(A,1)).
• Use Matrices.solve to solve the linear equation A*x = b, instead of using determinants to compute the solution.

See also

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

Inputs

Outputs

Modelica.Math.Matrices.inv

Return inverse of a matrix (try to avoid inv(..))

Information

Syntax

invA = Matrices.inv(A);

Description

This function returns the inverse of matrix A, i.e., A*inv(A) = identity(size(A,1)) computed by a LU decomposition with row pivoting. Usually, this function should not be used, because there are nearly always better numerical algorithms as by computing directly the inverse. Example:

Use x = Matrices.solve(A,b) to solve the linear equation A*x = b, instead of computing the solution by x = inv(A)*b, because this is much more efficient and much more reliable.

See also

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

Inputs

Outputs

Modelica.Math.Matrices.rank

Return rank of a rectangular matrix (computed with singular values)

Information

Syntax

result = Matrices.rank(A);
result = Matrices.rank(A,eps=0);

Description

This function returns the rank of a square or rectangular matrix A computed by singular value decomposition. For details about the rank of a matrix, see http://en.wikipedia.org/wiki/Matrix_rank. To be more precise:

• rank(A) returns the number of singular values of A that are larger than max(size(A))*norm(A)*Modelica.Constants.eps.
• rank(A, eps) returns the number of singular values of A that are larger than "eps".

See also

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

Inputs

Outputs

Modelica.Math.Matrices.conditionNumber

Return the condition number norm(A)*norm(inv(A)) of a matrix A

Information

Syntax

r = Matrices.conditionNumber(A);

Description

This function calculates the condition number (norm(A) * norm(inv(A))) of a general real matrix A, in either the 1-norm, 2-norm or the infinity-norm. In the case of 2-norm the result is the ratio of the largest to the smallest singular value of A. For more details, see http://en.wikipedia.org/wiki/Condition_number.

Example

A = [1, 2;
     2, 1];
r = conditionNumber(A);

results in:

r = 3.0

See also

Matrices.rcond

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

Inputs

Outputs

Modelica.Math.Matrices.rcond

Return the reciprocal condition number of a matrix

Information

Syntax

r = Matrices.rcond(A);

Description

This function estimates the reciprocal of the condition number (norm(A) * norm(inv(A))) of a general real matrix A, in either the 1-norm or the infinity-norm, using the LAPACK function DGECON. If rcond(A) is near 1.0, A is well conditioned and A is ill conditioned if rcond(A) is near zero.

Example

A = [1, 2;
     2, 1];
r = rcond(A);

results in:

r = 0.3333

See also

Matrices.conditionNumber

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

Inputs

Outputs

Modelica.Math.Matrices.norm

Return the p-norm of a matrix

Information

Syntax

Matrices.norm(A);
Matrices.norm(A, p=2);

Description

The function call "Matrices.norm(A)" returns the 2-norm of matrix A, i.e., the largest singular value of A.
The function call "Matrices.norm(A, p)" returns the p-norm of matrix A. The only allowed values for p are

• "p=1": the largest column sum of A
• "p=2": the largest singular value of A
• "p=Modelica.Constants.inf": the largest row sum of A

Note, for any matrices A1, A2 the following inequality holds:

Matrices.norm(A1+A2,p) ≤ Matrices.norm(A1,p) + Matrices.norm(A2,p)

Note, for any matrix A and vector v the following inequality holds:

Vectors.norm(A*v,p) ≤ Matrices.norm(A,p)*Vectors.norm(A,p)

See also

Matrices.frobeniusNorm

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

Inputs

Outputs

Modelica.Math.Matrices.frobeniusNorm

Return the Frobenius norm of a matrix

Information

Syntax

r = Matrices.frobeniusNorm(A);

Description

This function computes the Frobenius norm of a general real matrix A, i.e., the square root of the sum of the squares of all elements.

Example

A = [1, 2;
     2, 1];
r = frobeniusNorm(A);

results in:

r = 3.162;

See also

Matrices.norm

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

Inputs

Outputs

Modelica.Math.Matrices.nullSpace

Return the orthonormal nullspace of a matrix

Information

Syntax

           Z = Matrices.nullspace(A);
(Z, nullity) = Matrices.nullspace(A);

Description

This function calculates an orthonormal basis Z=[z_1, z_2, ...] of the nullspace of a matrix A, i.e., A*z_i=0.

The nullspace is obtained by SVD method. That is, matrix A is decomposed into the matrices S, U, V:

A = U*S*transpose(V)

with the orthonormal matrices U and V and the matrix S with

S = [S1, 0]
S1 = [diag(s); 0]

and the singular values s={s1, s2, ..., sr} of A and r=rank(A). Note, that S has the same size as A. Since U and V are orthonormal we may write

transpose(U)*A*V = [S1, 0].

Matrix S1 obviously has full column rank and therefore, the left n-r rows (n is the number of columns of A or S) of matrix V span a nullspace of A.

The nullity of matrix A is the dimension of the nullspace of A. In view of the above, it becomes clear that nullity holds

nullity = n - r

with

n = number of columns of matrix A
r = rank(A)

Example

A = [1, 2,  3, 1;
     3, 4,  5, 2;
    -1, 2, -3, 3];
(Z, nullity) = nullspace(A);

results in:

Z=[0.1715;
  -0.686;
   0.1715;
   0.686]

nullity = 1

See also

Matrices.singularValues

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

Inputs

Outputs

Modelica.Math.Matrices.exp

Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing

Information

Syntax

phi = Matrices.exp(A);
phi = Matrices.exp(A,T=1);

Description

This function computes the exponential e^AT of matrix A, i.e.

                       (AT)^2   (AT)^3
 Φ = e^(AT) = I + AT + ------ + ------ + ....
                         2!       3!

where e=2.71828..., A is an n x n matrix with real elements and T is a real number, e.g., the sampling time. A may be singular. With the exponential of a matrix it is, e.g., possible to compute the solution of a linear system of differential equations

der(x) = A*x   ->   x(t0 + T) = e^(AT)*x(t0)

Algorithmic details

The algorithm is taken from

H. D. Joos, G. Grübel:

RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

The following steps are performed to calculate the exponential of A:

1. Matrix A is balanced
   (= is transformed with a diagonal matrix D, such that inv(D)*A*D has a smaller condition as A).
2. The scalar T is divided by a multiple of 2 such that norm( inv(D)*A*D*T/2^k ) < 0.5. Note, that (1) and (2) are implemented such that no round-off errors are introduced.
3. The matrix from (2) is approximated by explicitly performing the Taylor series expansion with a variable number of terms. Truncation occurs if a new term does no longer contribute to the value of Φ from the previous iteration.
4. The resulting matrix is transformed back, by reverting the steps of (2) and (1).

In several sources it is not recommended to use Taylor series expansion to calculate the exponential of a matrix, such as in 'C.B. Moler and C.F. Van Loan: Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20, pp. 801-836, 1979' or in the documentation of m-file expm2 in MATLAB version 6 (http://www.mathworks.com) where it is stated that 'As a practical numerical method, this is often slow and inaccurate'. These statements are valid for a direct implementation of the Taylor series expansion, but not for the implementation variant used in this function.

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

Inputs

Outputs

Modelica.Math.Matrices.integralExp

Return the exponential and the integral of the exponential of a matrix

Information

Syntax

(phi,gamma) = Matrices.integralExp(A,B);
(phi,gamma) = Matrices.integralExp(A,B,T=1);

Description

This function computes the exponential phi = e^(AT) of matrix A and the integral gamma = integral(phi*dt)*B.

The function uses a Taylor series expansion with Balancing and scaling/squaring to approximate the integral Ψ of the matrix exponential Φ=e^(AT):

                         AT^2   A^2 * T^3          A^k * T^(k+1)
Ψ = int(e^(As))ds = IT + ---- + --------- + ... + --------------
                          2!        3!                (k+1)!

Φ is calculated through Φ = I + A*Ψ, so A may be singular. Γ is simply Ψ*B.

The algorithm runs in the following steps:

1. Balancing
2. Scaling
3. Taylor series expansion
4. Re-scaling
5. Re-Balancing

Balancing put the bad condition of a square matrix A into a diagonal transformation matrix D. This reduce the effort of following calculations. Afterwards the result have to be re-balanced by transformation D*Atransf *inv(D).
Scaling halfen T  k-times, until the norm of A*T is less than 0.5. This guarantees minimum rounding errors in the following series expansion. The re-scaling based on the equation  exp(A*2T) = exp(AT)^2. The needed re-scaling formula for psi thus becomes:

      Φ = Φ'*Φ'
I + A*Ψ = I + 2A*Ψ' + A^2*Ψ'^2
      Ψ = A*Ψ'^2 + 2*Ψ'

where psi' is the scaled result from the series expansion while psi is the re-scaled matrix.

The function is normally used to discretize a state-space system as the zero-order-hold equivalent:

x(k+1) = Φ*x(k) + Γ*u(k)
  y(k) = C*x(k) + D*u(k)

The zero-order-hold sampling, also known as step-invariant method, gives exact values of the state variables, under the assumption that the control signal u is constant between the sampling instants. Zero-order-hold sampling is described in

K. J. Åström, B. Wittenmark:

Computer Controlled Systems - Theory and Design
Third Edition, p. 32

Syntax:
      (phi,gamma) = Matrices.expIntegral(A,B,T)
                       A,phi: [n,n] square matrices
                     B,gamma: [n,m] input matrix
                           T: scalar, e.g., sampling time

The Algorithm to calculate psi is taken from

H. D. Joos, G. Grübel:

RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

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

Inputs

Outputs

Modelica.Math.Matrices.integralExpT

Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix

Information

(phi,gamma,gamma1) = Matrices.integralExpT(A,B);
(phi,gamma,gamma1) = Matrices.integralExpT(A,B,T=1);

Description

This function computes the exponential phi = e^(AT) of matrix A and the integral gamma = integral(phi*dt)*B and the integral integral((T-t)*exp(A*t)*dt)*B, where A is a square (n,n) matrix and B, gamma, and gamma1 are (n,m) matrices.

The function calculates the matrices phi,gamma,gamma1 through the equation:

                                 [ A B 0 ]
[phi gamma gamma1] = [I 0 0]*exp([ 0 0 I ]*T)
                                 [ 0 0 0 ]

The matrices define the discretized first-order-hold equivalent of a state-space system:

x(k+1) = phi*x(k) + gamma*u(k) + gamma1/T*(u(k+1) - u(k))

The first-order-hold sampling, also known as ramp-invariant method, gives more smooth control signals as the ZOH equivalent. First-order-hold sampling is, e.g., described in

K. J. Åström, B. Wittenmark:

Computer Controlled Systems - Theory and Design
Third Edition, p. 256

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

Inputs

Outputs

Modelica.Math.Matrices.continuousLyapunov

Return solution X of the continuous-time Lyapunov equation X*A + A'*X = C

Information

Syntax

X = Matrices.continuousLyapunov(A, C);
X = Matrices.continuousLyapunov(A, C, ATisSchur, eps);

Description

This function computes the solution X of the continuous-time Lyapunov equation

X*A + A'*X = C

using the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

Y*R' + R*Y = D

with R=U'*A'*U is the real Schur form of A' and D=U'*C*U and Y=U'*X*U are the corresponding transformations of C and X. This problem is solved sequentially for the 1x1 or 2x2 Schur blocks by exploiting the block triangular form of R. Finally the solution of the original problem is recovered as X=U*Y*U'.
The Boolean input "ATisSchur" indicates to omit the transformation to Schur in the case that A' has already Schur form.

References

[1] Bartels, R.H. and Stewart G.W.
    Algorithm 432: Solution of the matrix equation AX + XB = C.
    Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

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

C =  [-2, 3, 1, 0;
      -6, 8, 0, 1;
       2, 3, 4, 5;
      0, -2, 0, 0];

X = continuousLyapunov(A, C);

results in:

X = [1.633, -0.761,  0.575, -0.656;
    -1.158,  1.216,  0.047,  0.343;
    -1.066, -0.052, -0.916,  1.61;
    -2.473,  0.717, -0.986,  1.48]

See also

Matrices.continuousSylvester, Matrices.discreteLyapunov

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

Inputs

Outputs

Modelica.Math.Matrices.continuousSylvester

Return solution X of the continuous-time Sylvester equation A*X + X*B = C

Information

Syntax

X = Matrices.continuousSylvester(A, B, C);
X = Matrices.continuousSylvester(A, B, C, AisSchur, BisSchur);

Description

Function continuousSylvester computes the solution X of the continuous-time Sylvester equation

A*X + X*B = C.

using the Schur method for Sylvester equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

S*Y + Y*T = D.

with S=U'*A*U is the real Schur of A, T=V'*T*V is the real Schur form of B and D=U'*C*V and Y=U*X*V' are the corresponding transformations of C and X. This problem is solved sequentially by exploiting the block triangular form of S and T. Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisSchur" and "BisSchur" indicate to omit one or both of the transformation to Schur in the case that A and/or B have already Schur form.

The function applies LAPACK-routine DTRSYL. See LAPACK.dtrsyl for more information.

References

[1] Bartels, R.H. and Stewart G.W.
    Algorithm 432: Solution of the matrix equation AX + XB = C.
    Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

A = [17.0,   24.0,   1.0,   8.0,   15.0 ;
     23.0,    5.0,   7.0,  14.0,   16.0 ;
      0.0,    6.0,  13.0,  20.0,   22.0;
      0.0,    0.0,  19.0,  21.0,    3.0 ;
      0.0,    0.0,   0.0,   2.0,    9.0];

B =  [8.0, 1.0, 6.0;
      0.0, 5.0, 7.0;
      0.0, 9.0, 2.0];

C = [62.0,  -12.0, 26.0;
     59.0,  -10.0, 31.0;
     70.0,  -6.0,   9.0;
     35.0,  31.0,  -7.0;
     36.0, -15.0,   7.0];

X = continuousSylvester(A, B, C);

results in:

X = [0.0,  0.0,  1.0;
     1.0,  0.0,  0.0;
     0.0,  1.0,  0.0;
     1.0,  1.0, -1.0;
     2.0, -2.0,  1.0];

See also

Matrices.discreteSylvester, Matrices.continuousLyapunov

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

Inputs

Outputs

Modelica.Math.Matrices.continuousRiccati

Return solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0 (care)

Information

Syntax

                        X = Matrices.continuousRiccati(A, B, R, Q);
(X, alphaReal, alphaImag) = Matrices.continuousRiccati(A, B, R, Q, true);

Description

Function continuousRiccati computes the solution X of the continuous-time algebraic Riccati equation

A'*X + X*A - X*G*X + Q = 0

with G = B*inv(R)*B' using the Schur vector approach proposed by Laub [1].

It is assumed that Q is symmetric and positive semidefinite and R is symmetric, nonsingular and positive definite, (A,B) is stabilizable and (A,Q) is detectable.

These assumptions are not checked in this function !!

The assumptions guarantee that the Hamiltonian matrix

H = [A, -G; -Q, -A']

has no pure imaginary eigenvalue and can be put to an ordered real Schur form

U'*H*U = S = [S11, S12; 0, S22]

with orthogonal similarity transformation U. S is ordered in such a way, that S11 contains the n stable eigenvalues of the closed loop system with system matrix A - B*inv(R)*B'*X. If U is partitioned to

U = [U11, U12; U21, U22]

with dimensions according to S, the solution X is calculated by

X*U11 = U21.

With optional input refinement=true a subsequent iterative refinement based on Newton's method with exact line search is applied. See continuousRiccatiIterative for more information.

References

[1] Laub, A.J.
    A Schur Method for Solving Algebraic Riccati equations.
    IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.

Example

A = [0.0, 1.0;
     0.0, 0.0];

B = [0.0;
     1.0];

R = [1];

Q = [1.0, 0.0;
     0.0, 2.0];

X = continuousRiccati(A, B, R, Q);

results in:

X = [2.0, 1.0;
     1.0, 2.0];

See also

Matrices.Utilities.continuousRiccatiIterative, Matrices.discreteRiccati

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

Inputs

Outputs

Modelica.Math.Matrices.discreteLyapunov

Return solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C

Information

Syntax

X = Matrices.discreteLyapunov(A, C);
X = Matrices.discreteLyapunov(A, C, ATisSchur, sgn, eps);

Description

This function computes the solution X of the discrete-time Lyapunov equation

A'*X*A + sgn*X = C

where sgn=1 or sgn =-1. For sgn = -1, the discrete Lyapunov equation is a special case of the Stein equation:

A*X*B - X + Q = 0.

The algorithm uses the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

R*Y*R' + sgn*Y = D.

with R=U'*A'*U is the real Schur form of A' and D=U'*C*U and Y=U'*X*U are the corresponding transformations of C and X. This problem is solved sequentially by exploiting the block triangular form of R. Finally the solution of the original problem is recovered as X=U*Y*U'.
The Boolean input "ATisSchur" indicates to omit the transformation to Schur in the case that A' has already Schur form.

References

[1] Bartels, R.H. and Stewart G.W.
    Algorithm 432: Solution of the matrix equation AX + XB = C.
    Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

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

C =  [-2,  3, 1, 0;
      -6,  8, 0, 1;
       2,  3, 4, 5;
       0, -2, 0, 0];

X = discreteLyapunov(A, C, sgn=-1);

results in:

X  = [7.5735,   -3.1426,  2.7205, -2.5958;
     -2.6105,    1.2384, -0.9232,  0.9632;
      6.6090,   -2.6775,  2.6415, -2.6928;
     -0.3572,    0.2298,  0.0533, -0.27410];

See also

Matrices.discreteSylvester, Matrices.continuousLyapunov

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

Inputs

Outputs

Modelica.Math.Matrices.discreteSylvester

Return solution of the discrete-time Sylvester equation A*X*B + sgn*X = C

Information

Syntax

X = Matrices.discreteSylvester(A, B, C);
X = Matrices.discreteSylvester(A, B, C, AisHess, BTisSchur, sgn, eps);

Description

Function discreteSylvester computes the solution X of the discrete-time Sylvester equation

A*X*B + sgn*X = C.

where sgn = 1 or sgn = -1. The algorithm applies the Hessenberg-Schur method proposed by Golub et al [1]. For sgn = -1, the discrete Sylvester equation is also known as Stein equation:

A*X*B - X + Q = 0.

In a nutshell, the problem is reduced to the corresponding problem

H*Y*S' + sgn*Y = F.

with H=U'*A*U is the Hessenberg form of A and S=V'*B'*V is the real Schur form of B', F=U'*C*V and Y=U*X*V' are appropriate transformations of C and X. This problem is solved sequentially by exploiting the specific forms of S and H. Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisHess" and "BTisSchur" indicate to omit one or both of the transformation to Hessenberg form or Schur form respectively in the case that A and/or B have already Hessenberg form or Schur respectively.

References

[1] Golub, G.H., Nash, S. and Van Loan, C.F.
    A Hessenberg-Schur method for the problem AX + XB = C.
    IEEE Transaction on Automatic Control, AC-24, no. 6, pp. 909-913, 1979.

Example

A = [1.0,   2.0,   3.0;
     6.0,   7.0,   8.0;
     9.0,   2.0,   3.0];

B = [7.0,   2.0,   3.0;
     2.0,   1.0,   2.0;
     3.0,   4.0,   1.0];

C = [271.0,   135.0,   147.0;
     923.0,   494.0,   482.0;
     578.0,   383.0,   287.0];

X = discreteSylvester(A, B, C);

results in:
X = [2.0,   3.0,   6.0;
     4.0,   7.0,   1.0;
     5.0,   3.0,   2.0];

See also

Matrices.continuousSylvester, Matrices.discreteLyapunov

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

Inputs

Outputs

Modelica.Math.Matrices.discreteRiccati

Return solution of discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0 (dare)

Information

Syntax

                        X = Matrices.discreteRiccati(A, B, R, Q);
(X, alphaReal, alphaImag) = Matrices.discreteRiccati(A, B, R, Q, true);

Description

Function discreteRiccati computes the solution X of the discrete-time algebraic Riccati equation

A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

using the Schur vector approach proposed by Laub [1].

It is assumed that Q is symmetric and positive semidefinite and R is symmetric, nonsingular and positive definite, (A,B) is stabilizable and (A,Q) is detectable. Using this method, A has also to be invertible.

These assumptions are not checked in this function !!!

The assumptions guarantee that the Hamiltonian matrix.

H = [A + G*T*Q, -G*T; -T*Q, T]

with

     -T
T = A

and

       -1
G = B*R *B'

has no eigenvalue on the unit circle and can be put to an ordered real Schur form

U'*H*U = S = [S11, S12; 0, S22]

with orthogonal similarity transformation U. S is ordered in such a way, that S11 contains the n stable eigenvalues of the closed loop system with system matrix

                  -1
A - B*(R + B'*X*B)  *B'*X*A

If U is partitioned to

U = [U11, U12; U21, U22]

according to S, the solution X can be calculated by

X*U11 = U21.

References

[1] Laub, A.J.
    A Schur Method for Solving Algebraic Riccati equations.
    IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.

Example

A  = [4.0    3.0]
     -4.5,  -3.5];

B  = [ 1.0;
      -1.0];

R = [1.0];

Q = [9.0, 6.0;
     6.0, 4.0]

X = discreteRiccati(A, B, R, Q);

  results in:

X = [14.5623, 9.7082;
      9.7082, 6.4721];

See also

Matrices.continuousRiccati

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

Inputs

Outputs

Modelica.Math.Matrices.sort

Sort the rows or columns of a matrix in ascending or descending order

Information

Syntax

           sorted_M = Matrices.sort(M);
(sorted_M, indices) = Matrices.sort(M, sortRows=true, ascending=true);

Description

Function sort(..) sorts the rows of a Real matrix M in ascending order and returns the result in sorted_M. If the optional argument "sortRows" is false, the columns of the matrix are sorted. If the optional argument "ascending" is false, the rows or columns are sorted in descending order. In the optional second output argument, the indices of the sorted rows or columns with respect to the original matrix are given, such that

sorted_M = if sortedRow then M[indices,:] else M[:,indices];

Example

(M2, i2) := Matrices.sort([2, 1,  0;
                           2, 0, -1]);
     -> M2 = [2, 0, -1;
              2, 1, 0 ];
        i2 = {2,1};

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

Inputs

Outputs

Modelica.Math.Matrices.flipLeftRight

Flip the columns of a matrix in left/right direction

Information

Syntax

A_flr = Matrices.flipLeftRight(A);

Description

Function flipLeftRight computes from matrix A a matrix A_flr with flipped columns, i.e., A_flr[:,i]=A[:,n-i+1], i=1,..., n.

Example

A = [1, 2,  3;
     3, 4,  5;
    -1, 2, -3];

A_flr = flipLeftRight(A);

results in:

A_flr = [3, 2,  1;
         5, 4,  3;
        -3, 2, -1]

See also

Matrices.flipUpDown

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

Inputs

Outputs

Modelica.Math.Matrices.flipUpDown

Flip the rows of a matrix in up/down direction

Information

Syntax

A_fud = Matrices.flipUpDown(A);

Description

Function flipUpDown computes from matrix A a matrix A_fud with flipped rows, i.e., A_fud[i,:]=A[n-i+1,:], i=1,..., n.

Example

A = [1, 2,  3;
     3, 4,  5;
    -1, 2, -3];

A_fud = flipUpDown(A);

results in:

A_fud  = [-1, 2, -3;
           3, 4,  5;
           1, 2,  3]

See also

Matrices.flipLeftRight

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

Inputs

Outputs