The Reduction to Hamiltonian Schur Form Explained David S. Watkins - - PowerPoint PPT Presentation

The Reduction to Hamiltonian Schur Form Explained

David S. Watkins

watkins@math.wsu.edu

Department of Mathematics Washington State University

July 2006 – p.1

Linear-Quadratic Gaussian Problem

July 2006 – p.2

Linear-Quadratic Gaussian Problem

Optimal control theory

July 2006 – p.2

Linear-Quadratic Gaussian Problem

Optimal control theory Solve algebraic Riccati equation

July 2006 – p.2

Linear-Quadratic Gaussian Problem

Optimal control theory Solve algebraic Riccati equation Find stable invariant subspace of a Hamiltonian matrix

July 2006 – p.2

Linear-Quadratic Gaussian Problem

Optimal control theory Solve algebraic Riccati equation Find stable invariant subspace of a Hamiltonian matrix Compute Hamiltonian Schur form (Paige/Van Loan 81)

July 2006 – p.2

Linear-Quadratic Gaussian Problem

Optimal control theory Solve algebraic Riccati equation Find stable invariant subspace of a Hamiltonian matrix Compute Hamiltonian Schur form (Paige/Van Loan 81) Chu/Liu/Mehrmann 2004

July 2006 – p.2

Hamiltonian Matrices

July 2006 – p.3

Hamiltonian Matrices

H =

• A

N K −AT

• ∈ R2n×2n

NT = N KT = K

July 2006 – p.3

Hamiltonian Matrices

H =

• A

N K −AT

• ∈ R2n×2n

NT = N KT = K

Hamiltonian spectrum

July 2006 – p.3

Hamiltonian Matrices

H =

• A

N K −AT

• ∈ R2n×2n

NT = N KT = K

Hamiltonian spectrum assume no purely imaginary eigenvalues

July 2006 – p.3

Hamiltonian Matrices

H =

• A

N K −AT

• ∈ R2n×2n

NT = N KT = K

Hamiltonian spectrum assume no purely imaginary eigenvalues stable invariant subspace

July 2006 – p.3

Hamiltonian Matrices, continued

July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• T is quasitriangular

July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• T is quasitriangular

eigenvalues of T in left half plane

July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• T is quasitriangular

eigenvalues of T in left half plane transforming matrix orthogonal and symplectic

July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• T is quasitriangular

eigenvalues of T in left half plane transforming matrix orthogonal and symplectic This gives the stable invariant subspace.

July 2006 – p.4

Hamiltonian Matrices, continued

Hamiltonian Schur form:

• T

N −T T

• T is quasitriangular

eigenvalues of T in left half plane transforming matrix orthogonal and symplectic This gives the stable invariant subspace. How can we compute it?

July 2006 – p.4

Skew-Hamiltonian Matrices

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• B is quasitriangular

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• B is quasitriangular

easier than Hamiltonian

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• B is quasitriangular

easier than Hamiltonian want to avoid squaring

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• B is quasitriangular

easier than Hamiltonian want to avoid squaring symplectic URV decomposition

July 2006 – p.5

Skew-Hamiltonian Matrices

H Hamiltonian ⇒ H2 skew Hamiltonian

skew-Hamiltonian Schur form:

• B

M BT

• B is quasitriangular

easier than Hamiltonian want to avoid squaring symplectic URV decomposition

H =

• A

N K −AT

• and

H2 =

• B

M BT

• July 2006 – p.5

Assumptions

July 2006 – p.6

Assumptions

all eigenvalues of H are real (time)

July 2006 – p.6

Assumptions

all eigenvalues of H are real (time)

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.6

Assumptions

all eigenvalues of H are real (time)

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

nongeneric cases ignored

July 2006 – p.6

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.7

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗           e1 is eigenvector of H2 but not of H

July 2006 – p.7

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗           e1 is eigenvector of H2 but not of H

span{e1, He1} is invariant under H

July 2006 – p.7

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗           e1 is eigenvector of H2 but not of H

span{e1, He1} is invariant under H

eigenvalues: ±λ1

July 2006 – p.7

H2 =           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗           e1 is eigenvector of H2 but not of H

span{e1, He1} is invariant under H

eigenvalues: ±λ1 Extract eigenvector x with eigenvalue λ1 < 0.

July 2006 – p.7

Use eigenvector x.

July 2006 – p.8

Use eigenvector x. Build orthogonal, symplectic transforming matrix with first column proportional to x.

July 2006 – p.8

Use eigenvector x. Build orthogonal, symplectic transforming matrix with first column proportional to x.

Qe1 = αx QT x = α−1e1

July 2006 – p.8

Use eigenvector x. Build orthogonal, symplectic transforming matrix with first column proportional to x.

Qe1 = αx QT x = α−1e1 ˆ H = QT HQ

July 2006 – p.8

Use eigenvector x. Build orthogonal, symplectic transforming matrix with first column proportional to x.

Qe1 = αx QT x = α−1e1 ˆ H = QT HQ ˆ H =           λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.8

Use eigenvector x. Build orthogonal, symplectic transforming matrix with first column proportional to x.

Qe1 = αx QT x = α−1e1 ˆ H = QT HQ ˆ H =           λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

Deflate?

July 2006 – p.8

A Difficulty

July 2006 – p.9

A Difficulty

What is the form of ˆ

H

2?

July 2006 – p.9

A Difficulty

What is the form of ˆ

H

2?

Want ˆ

H

2 =

          ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.9

A Difficulty

What is the form of ˆ

H

2?

Want ˆ

H

2 =

          ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗           Q must be built with care.

July 2006 – p.9

Construction of QT

July 2006 – p.10

Construction of QT

x =           ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒           ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒           ∗ ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒           ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗ ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗          

July 2006 – p.10

Construction of QT

⇒ ⇒ ⇒ ⇒           ∗          

Done!

July 2006 – p.10

Why it Works

July 2006 – p.11

Why it Works

H2 stays in skew-Hamiltonian Schur form . . .

July 2006 – p.11

Why it Works

H2 stays in skew-Hamiltonian Schur form . . .

. . . every step of the way.

July 2006 – p.11

Why it Works

H2 stays in skew-Hamiltonian Schur form . . .

. . . every step of the way.

Hx = λ1x

July 2006 – p.11

Why it Works

H2 stays in skew-Hamiltonian Schur form . . .

. . . every step of the way.

Hx = λ1x H2x = λ2

1x

July 2006 – p.11

Why it Works

H2 stays in skew-Hamiltonian Schur form . . .

. . . every step of the way.

Hx = λ1x H2x = λ2

1x

relationships preserved throughout the transformation

July 2006 – p.11

Close up

July 2006 – p.12

Close up

H2x = xλ2

1

July 2006 – p.12

Close up

H2x = xλ2

1

          λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗ ∗ ∗ ∗ ∗ ∗           =           ∗ ∗ ∗ ∗ ∗ ∗           λ2

1

July 2006 – p.12

Close up

H2x = xλ2

1

          λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗ ∗ ∗ ∗ ∗ ∗           =           ∗ ∗ ∗ ∗ ∗ ∗           λ2

1

July 2006 – p.12

   λ2

1

∗ λ2

2

∗ ∗ λ2

3

      ∗ ∗ ∗    =    ∗ ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

1

∗ λ2

2

∗ ∗ λ2

3

      ∗ ∗ ∗    =    ∗ ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    ∗ ∗ ∗ ∗ ∗ ∗ λ2

3

      ∗ ∗    =    ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    ∗ ∗ ∗ ∗ ∗ λ2

3

      ∗ ∗    =    ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    ∗ ∗ λ2

1

∗ ∗ λ2

3

      ∗ ∗    =    ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ λ2

1

∗ ∗ λ2

3

      ∗ ∗    =    ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ λ2

1

∗ ∗ λ2

3

      ∗ ∗    =    ∗ ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ ∗ ∗ ∗ ∗ ∗       ∗    =    ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ ∗ ∗ ∗ ∗       ∗    =    ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ ∗ ∗ ∗ λ2

1

      ∗    =    ∗    λ2

1

July 2006 – p.13

⇒ ⇒    λ2

2

∗ λ2

3

∗ ∗ λ2

1

      ∗    =    ∗    λ2

1

July 2006 – p.13

   λ2

2

∗ λ2

3

∗ ∗ λ2

1

      ∗    =    ∗    λ2

1

July 2006 – p.13

          λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ λ2

1

                    ∗ ∗ ∗ ∗           =           ∗ ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ λ2

1

                    ∗ ∗ ∗ ∗           =           ∗ ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ λ2

1

                    ∗ ∗ ∗ ∗           =           ∗ ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ λ2

1

                    ∗ ∗ ∗           =           ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ λ2

1

                    ∗ ∗ ∗           =           ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

3

∗ ∗ ∗ ∗ λ2

1

∗ ∗ λ2

2

∗ λ2

3

∗ ∗ ∗ λ2

1

                    ∗ ∗ ∗           =           ∗ ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

1

∗ ∗ ∗ λ2

3

∗ ∗ λ2

2

∗ λ2

1

∗ ∗ λ2

3

                    ∗ ∗           =           ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ λ2

1

∗ ∗ ∗ λ2

3

∗ ∗ λ2

2

∗ λ2

1

∗ ∗ λ2

3

                    ∗ ∗           =           ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

2

∗ ∗ ∗ ∗ ∗ λ2

1

∗ ∗ ∗ λ2

3

∗ ∗ λ2

2

∗ ∗ λ2

1

∗ ∗ λ2

3

                    ∗ ∗           =           ∗ ∗           λ2

1

July 2006 – p.14

⇒ ⇒ ⇒ ⇒           λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗           =           ∗           λ2

1

July 2006 – p.14

          λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗           =           ∗           λ2

1

July 2006 – p.14

          λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗           =           ∗           λ2

1

It looks like we’re back where we started,

July 2006 – p.15

          λ2

1

∗ ∗ ∗ ∗ λ2

2

∗ ∗ ∗ λ2

3

∗ ∗ λ2

1

∗ λ2

2

∗ ∗ λ2

3

                    ∗           =           ∗           λ2

1

It looks like we’re back where we started, but . . .

July 2006 – p.15

H =           λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

July 2006 – p.16

H =           λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗          

Now we can deflate.

July 2006 – p.16

In Conclusion

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form every step of the way

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form every step of the way we can transform H to Hamiltonian Schur form.

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form every step of the way we can transform H to Hamiltonian Schur form. This work will appear in ETNA.

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form every step of the way we can transform H to Hamiltonian Schur form. This work will appear in ETNA. Download from my website. (Google David S Watkins)

July 2006 – p.17

In Conclusion

By keeping H2 in skew-Hamiltonian Schur form every step of the way we can transform H to Hamiltonian Schur form. This work will appear in ETNA. Download from my website. (Google David S Watkins)

Thank you for your attention.

July 2006 – p.17