- Lyapunov equation
In
control theory , the discrete Lyapunov equation is of the form:A X A^H - X + Q = 0where Q is ahermitian matrix . The continuous Lyapunov equation is of form:AX + XA^H + Q = 0.The Lyapunov equation occurs in many branches of control theory, such as stability analysis and
optimal control . This and related equations are named after the Russian mathematicianAleksandr Lyapunov .Application to stability
In the following theorems A, P, Q in mathbb{R}^{n imes n}, and P and Q are symmetric. The notation P>0 means that the matrix P is positive definite
Theorem (continuous time version). If there exist P>0 and Q>0 satisfying A^T P + P A + Q = 0 then the linear system dot{x}=A x is globally asymptotically stable. The quadratic function V(z)=z^T P z is a
Lyapunov function that can be used to verify stability.Theorem (discrete time version). If there exist P>0 and Q>0 satisfying A^T P A -P + Q = 0 then the linear system x(t+1)=A x(t) is globally asymptotically stable. As before, z^T P z is a Lyapunov function.
Computational aspects of solution
The discrete Lyapunov equations can, by using
Schur complement s, be written as:egin{bmatrix}X^{-1} & A \ A^H & X-Qend{bmatrix}=0or equivalently as:egin{bmatrix}X & XA \ A^HX & X-Qend{bmatrix}=0.Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa (1977) is often used. For the continuous Lyapunov equation the method of Bartels and Stewart (1972) can be used.
ee also
*
Sylvester equation References
* Kitagawa: "An Algorithm for Solving the Matrix Equation X = F X F' + S", International Journal of Control, Vol. 25, No. 5, p745–753 (1977).
* R. H. Bartels and G. W. Stewart: "Algorithm 432: Solution of the matrix equation AX + XB = C", Comm. ACM, 15 (1972), p820-826.
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