Solve a discrete algebraic Riccati equation:

Verify the solution:

Solve a discrete Riccati equation:

Solve a discrete Riccati equation with a cross-coupling matrix:

Solve the Riccati equation for a sampled-data transfer-function model:

The linearized model of a turbo-generator has state and input matrices:

The discrete LQ regulation cost J_opt for given weighting matrices {q,r} and initial state s₀:

Compute an optimal state-feedback gain that guarantees that all the closed-loop poles are inside a circle of radius :

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

DiscreteRiccatiSolve[{a,b},{q,r,p}] is equivalent to DiscreteRiccatiSolve[{a-b..p,b},{q-p.r ^-1.p,r}]:

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

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

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

The matrix associated with the discrete algebraic Riccati equation is symplectic:

The eigenvalues of the symplectic matrix are pairs of the form :

and are similar matrices:

The symplectic matrix must satisfy the stability and complementarity properties 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 symplectic matrix:

Compute optimal state-feedback gains using DiscreteRiccatiSolve:

Use LQRegulatorGains to obtain the same result directly:

Compute optimal output feedback gains using DiscreteRiccatiSolve:

LQOutputRegulatorGains gives the same result:

Compute optimal estimator gains using DiscreteRiccatiSolve:

Use LQEstimatorGains:

The optimal trajectory of the discrete approximation of a system results in a higher cost:

The optimal LQ regulation costs for the systems starting at s₀:

If {a,b} is not stabilizable and {a,g} is not detectable, then the Riccati equation with q=g.g has no stabilizing solution: