Solve a continuous algebraic Riccati equation:

Verify the solution:

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:

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

The optimal cost is .x.s_init:

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:

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:

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