- Application to stability
- Computational aspects of solution
- Analytic solution Discrete time Continuous time
- See also
- References
In control theory, the discrete Lyapunov equation is of the form
where 
is a Hermitian matrix and 
is the conjugate transpose of 
.
The continuous Lyapunov equation is of form: 
.
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 mathematician Aleksandr Lyapunov.
Application to stability
In the following theorems 
, and 
and are symmetric. The notation 
means that the matrix is positive definite.
Theorem (continuous time version). Given any 
, there exists a unique satisfying 
if and only if the linear system 
is globally asymptotically stable. The quadratic function 
is a Lyapunov function that can be used to verify stability.
Theorem (discrete time version). Given any , there exists a unique satisfying 
if and only if the linear system 
is globally asymptotically stable. As before, 
is a Lyapunov function.
Computational aspects of solution
Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used.[1] For the continuous Lyapunov equation the method of Bartels and Stewart can be used.[2]
Analytic solution
Defining the 
operator as stacking the columns of a matrix and 
as the Kronecker product of and 
, the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix is stable, the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).
Discrete time
Using the result that 
, one has
where 
is a conformable identity matrix.[3] One may then solve for 
by inverting or solving the linear equations. To get 
, one must just reshape 
appropriately.
Moreover, if is stable, the solution can also be written as
.
For comparison, consider the one-dimensional case, where this just says that the solution of 
is 
.
Continuous time
Using again the Kronecker product notation and the vectorization operator, one has the matrix equation
where 
denotes the matrix obtained by complex conjugating the entries of .
Similar to the discrete-time case, if is stable, the solution can also be written as
.
For comparison, consider the one-dimensional case, where this just says that the solution of 
is 
.
See also
- Sylvester equation
- Algebraic Riccati equation
- Kalman filter
References
2. ^{{cite journal |first=R. H. |last=Bartels |first2=G. W. |last2=Stewart |title=Algorithm 432: Solution of the matrix equation AX + XB = C |journal=Comm. ACM |volume=15 |year=1972 |issue=9 |pages=820–826 |doi=10.1145/361573.361582 }}
3. ^{{cite book |first=J. |last=Hamilton |year=1994 |title=Time Series Analysis |at=Equations 10.2.13 and 10.2.18 |location= |publisher=Princeton University Press |isbn=0-691-04289-6 }}
- Russian Alphabet
- Russian alphabet reforms
- Russian Alsos
- Russian-America
- Russian America Company
- Russian-American
- Russian American Gallium Experiment
- Russian American Medical Association
- Russian american medical association
- Russian Americans
- Russian-Americans
- Russian American Telegraph
- Russian Amerika
- Russia naming issue
- Russian anarchism
- Russian ancestry of Catherine the Great
- Russian annexation of Crimea
- Russian Arborvitae
- Russian archaeologist
- Russian Archaeology Institute
- Russian Archeological Congress
- Russian Architecture
- Russian aristocracy
- Russian Armenia
- Russian Armenians