ControlMatrixEquations.jl

Solvers for Sylvester, Lyapunov and Riccati Equations

Numerous forms of the matrix equations below (in terms of symbols, signs, and transposes) occur in the literature and other software packages. The ones used for this package were chosen to be sensible in a control setting, but is WIP.

Linear matrix equations

The packages provides direct solvers based a vanilla implementation of Bartels–Stewart's algorithm. If there is no method schur for the A or the B matrix, there is a fallback to the "naive" (Kronecker product) method, this is useful for, e.g., symbolic equations.

Sylvester Equations

• sylvc(A, B, C) solves AX + BX = C
• sylvd(A, B, C) solves AXB - X = C
• sylvg(A, B, C, E, F) solves AXB + EXF = C

Lyapunov Equations

• lyapc(A, Q) solves AX + XA' = -Q
• lyapd(A, Q) solves AXA' - X = -Q
• lyapc(A, Q, E) solves AXE' + EXA' = -Q (will possibly be changed to to lyapcg(E, A, Q))
• lyapd(A, Q, E) solves AXA' - EXE' = -Q

Riccati Equations

Schur-factorization based Riccati solvers, including extended pencil versions that handle singular and near singular R matrices.

• arec(A, B, Q, R, S) solves the equation A'X + XA - (XB + S)/R(XB + S)' + Q = 0
• ared(A, B, Q, R, S) solves the equation A'XA - X - (A'XB + S)/(B'XB + R)(A'XB + S)' + Q = 0
• arecg(E, A, B, Q, R, S) solves the equation A'XE + E'XA - (E'XB + S)/R(E'XB + S)' + Q = 0
• aredg(E, A, B, Q, R, S) solves the equation A'XA - E'XE - (A'XB + S)/(B'XB + R)(A'XB + S)' + Q = 0

See Arnold & Laub (1984) "Generalized eigenproblem algorithms and software for algebraic Riccati equations."