RiccatiSolve

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:

The eigenvalue decomposition method can be used for both exact and inexact systems :

For inexact systems, Schur decomposition may also be used:

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

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 guarantees that all closed-loop poles are to the left of :

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

Compute the minimum error covariance for a Kalman estimator:

RiccatiSolve is equivalent to RiccatiSolve {a-b.Inverse[r].ConjugateTranspose[p], b}, {q-p.Inverse[r].ConjugateTranspose[p], r}]:

If is stabilizable and is detectable, and q=Transpose[g].g, then the solution to the Riccati equation is positive semidefinite:

If is controllable and 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 :

and are similar matrices:

The Hamiltonian matrix must satisfy the stability and complementarity property to obtain a stabilizing solution to the Riccati equation:

Stability property:

Complementarity property:

The stabilizing solution:

The eigenvalues of the feedback system with are 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:

If is not stabilizable and is not detectable, then the Riccati equation with has no stabilizing solution:

The Schur decomposition method works only with approximate numerical matrices: