RiccatiSolve

RiccatiSolve[{a,b},{q,r}]

gives the matrix that is the stabilizing solution of the continuous algebraic Riccati equation TemplateBox[{a}, ConjugateTranspose].x+x.a-x.b.TemplateBox[{r}, Inverse].TemplateBox[{b}, ConjugateTranspose].x+q=0.

RiccatiSolve[{a,b},{q,r,p}]

solves the equation TemplateBox[{a}, ConjugateTranspose].x+x.a-(x.b+p).TemplateBox[{r}, Inverse].(TemplateBox[{b}, ConjugateTranspose].x+TemplateBox[{p}, ConjugateTranspose])+q=0.

Details and Options

  • In TemplateBox[{a}, ConjugateTranspose].x+x.a-x.b.TemplateBox[{r}, Inverse].TemplateBox[{b}, ConjugateTranspose].x+q=0, denotes the conjugate transpose.
  • The equation TemplateBox[{a}, ConjugateTranspose].x+x.a-x.b.TemplateBox[{r}, Inverse].TemplateBox[{b}, ConjugateTranspose].x+q=0 has a unique, symmetric, positive semidefinite solution if is stabilizable, is detectable, , and . Consequently, all eigenvalues of the matrix are negative and the solution is stabilizing.
  • The solution is positive definite when is controllable and is observable.
  • RiccatiSolve supports a Method option, with the following possible settings:
  • Automaticautomatically determined method
    "Eigensystem"based on eigen decomposition
    "GeneralizedEigensystem"based on generalized eigen decomposition
    "GeneralizedSchur"based on generalized Schur decomposition
    "InverseFree"a variant of "GeneralizedSchur"
    "MatrixSign"iterative method using the matrix sign function
    "Newton"iterative Newton method
    "Schur"based on Schur decomposition
  • All methods apply to approximate numeric matrices. "Eigensystem" also applies to exact and symbolic matrices.

Examples

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Basic Examples  (1)

Solve a continuous algebraic Riccati equation:

Verify the solution:

Scope  (3)

Solve a continuous Riccati equation:

Solve a Riccati equation with state-control coupling:

Solve the Riccati equation by extracting appropriate matrices from a state-space model object:

Options  (7)

Method  (7)

Automatic and "Eigensystem" methods can be used for exact systems:

As well as for inexact systems:

"Schur" method can be used for inexact systems:

"Newton" applies to inexact systems and may be more accurate than Automatic:

Compute the solution and absolute error:

Compare to the default method:

"Newton" takes suboptions "StartingMatrix", "MaxIterations", and "Tolerance":

The "Newton" method may not converge even if a stabilizing solution exists:

Compare with the Automatic method:

"MatrixSign" is typically used as an initial approximation for the "Newton" method:

Use xinit to initialize the "Newton" method:

Compare errors:

"MatrixSign" takes suboptions "MaxIterations" and "Tolerance":

"GeneralizedSchur" and "GeneralizedEigensystem" applies when a is singular:

The matrix a is singular:

These two methods work for a singular a:

Verify error:

"InverseFree" can be used when r is ill-conditioned:

The matrix r has a high condition number:

Use the method:

Compare to the default method:

The absolute error is higher for the default method in this case:

Applications  (3)

Compute the cost associated with an optimal trajectory for a linearized inverted pendulum:

The optimal cost is .x.sinit:

Obtain the cost by numerical integration:

Track the "cost to go" at each instant along the optimal trajectory:

Compute an optimal state-feedback gain that places all closed-loop poles to the left of :

The closed-loop poles without any prescribed degree of stability:

Compute the minimum error covariance for a Kalman estimator:

Properties & Relations  (8)

Find the optimal gains with cross-coupling matrix p:

An equivalent result can be found be incorporating p into the a and q matrices:

If {a,b} is controllable and {a,g} is observable, and q=Transpose[g].g, then the solution to the Riccati equation is positive definite:

The eigenvalues of the Hamiltonian matrix are pairs of the form {λ,-λ}:

and are similar matrices:

The ability to obtain a stabilizing solution depends on the Hamiltonian matrix:

The Hamiltonian matrix must satisfy the stability property:

As well as the complementarity property:

The stabilizing solution:

Find the eigenvalues of the feedback system with :

These are also the stable eigenvalues of the Hamiltonian matrix:

Compute optimal state feedback gains using RiccatiSolve:

Use LQRegulatorGains to compute the same result directly:

Compute optimal output feedback gains using RiccatiSolve:

LQOutputRegulatorGains gives the same result:

Compute optimal estimator gains using RiccatiSolve:

Use LQEstimatorGains to compute this result directly:

Possible Issues  (1)

If is not stabilizable or is not detectable, then the Riccati equation with q=TemplateBox[{g}, Transpose].g has no stabilizing solution:

Introduced in 2010
 (8.0)
 |
Updated in 2014
 (10.0)