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DiscreteRiccatiSolve

DiscreteRiccatiSolve
gives the matrix that is the stabilizing solution of the discrete algebraic Riccati equation .
DiscreteRiccatiSolve
solves .
  • In , denotes the conjugate transpose.
  • The equation has a unique, symmetric, positive semidefinite solution only if is stabilizable, is detectable, , and . Consequently, all the eigenvalues of the matrix lie inside the unit circle and the solution is stabilizing.
  • The solution is positive definite when is controllable and is observable.
  • The eigenvalues of the symplectic matrix must not contain any symbolic expressions.
"Eigensystem"uses eigenvalue decomposition
"Schur"uses Schur decomposition
  • The default setting Method->Automatic selects for exact matrices and as the primary method for real matrices.
  • Method only works with real matrices.
Solve a discrete algebraic Riccati equation:
Verify the solution:
Solve a discrete algebraic Riccati equation:
In[1]:=
Click for copyable input
Out[1]//MatrixForm=
Verify the solution:
In[2]:=
Click for copyable input
Out[2]=
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:
For exact systems, eigenvalue decomposition is used to obtain the solution:
For inexact systems, Schur decomposition is used:
Eigenvalue decomposition may also be used:
Compute the cost associated with an optimal trajectory for a typical linearized turbo-generator model:
Compute an optimal state-feedback gain which guarantees that all the closed-loop poles are inside a circle of radius :
The closed-loop poles without any prescribed degree of stability:
If is stabilizable and is detectable, and q=Transpose[g].g, then the solution to the discrete Riccati equation is positive semidefinite:
 
If is controllable and 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:
The optimal trajectory of the discrete approximation of a system results in a higher cost:
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:
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