Math 2.0

Linear and multilinear algebra; matrix theory Definition

Algebraic Riccati equation

Builds on MatrixSymmetric matrixPositive semidefinite matrixPositive definite matrixTranspose

Definition

Let n,m≥1n, m \ge 1. Let AA be a real n×nn \times n matrix and BB a real n×mn \times m matrix. Let QQ be a real symmetric positive semidefinite n×nn \times n matrix. Let RR be a real symmetric positive definite m×mm \times m matrix. Write MTM^T for the transpose of MM.

  1. The continuous-time algebraic Riccati equation is ATX+XA−XBR−1BTX+Q=0A^T X + XA - XBR^{-1}B^T X + Q = 0 for an unknown real symmetric n×nn \times n matrix XX.
  2. The discrete-time algebraic Riccati equation is X=ATXA−ATXB(R+BTXB)−1BTXA+QX = A^T X A - A^T X B (R + B^T X B)^{-1} B^T X A + Q for an unknown real symmetric positive semidefinite n×nn \times n matrix XX.

Remarks

A solution XX of the continuous equation is stabilising if every eigenvalue of A−BR−1BTXA - BR^{-1}B^T X has negative real part. If the pair (A,B)(A, B) is stabilisable and the pair (Q1/2,A)(Q^{1/2}, A) is detectable, the continuous equation has a unique positive semidefinite solution, and it is stabilising. The optimal feedback of the linear-quadratic regulator is then u=−R−1BTXxu = -R^{-1}B^T X x. Example: for n=m=1n = m = 1, A=0A = 0 and B=Q=R=1B = Q = R = 1, the continuous equation is 1−X2=01 - X^2 = 0, with solutions X=1X = 1 and X=−1X = -1. Only X=1X = 1 is stabilising, since A−BR−1BTX=−XA - BR^{-1}B^T X = -X. For B=0B = 0 the continuous equation becomes the Lyapunov equation ATX+XA+Q=0A^T X + XA + Q = 0, which is linear and not of Riccati type.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_Riccati_equation
About this page
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URIm2://APX4ZA01DK
Kinddefinition
WritingW0 stub
MSC15A24, 49N10, 93C05
DefinesAlgebraic Riccati equation
UsesMatrix, Symmetric matrix, Positive semidefinite matrix, Positive definite matrix, Transpose
Changed2 November 2026

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