Lyapunov equation

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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.[1][2]

Application to stability[edit]

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[edit]

Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used.[3] For the continuous Lyapunov equation the method of Bartels and Stewart can be used.[4]

Analytic solution[edit]

Defining the vectorization 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[edit]

Using the result that , one has

where is a conformable identity matrix and is the element-wise complex conjugate of .[5] 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[edit]

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 .

Relationship Between Discrete and Continuous Lyapunov Equations[edit]

We start with the continuous-time linear dynamics:

.

And then discretize it as follows:

Where indicates a small forward displacement in time. Substituting the bottom equation into the top and shuffling terms around, we get a discrete-time equation for .

Where we've defined . Now we can use the discrete time Lyapunov equation for :

Plugging in our definition for , we get:

Expanding this expression out yields:

Recall that is a small displacement in time. Letting go to zero brings us closer and closer to having continuous dynamics—and in the limit we achieve them. It stands to reason that we should also recover the continuous-time Lyapunov equations in the limit as well. Dividing through by on both sides, and then letting we find that:

which is the continuous-time Lyapunov equation, as desired.

See also[edit]

References[edit]

  1. ^ Parks, P. C. (1992-01-01). "A. M. Lyapunov's stability theory—100 years on *". IMA Journal of Mathematical Control and Information. 9 (4): 275–303. doi:10.1093/imamci/9.4.275. ISSN 0265-0754.
  2. ^ Simoncini, V. (2016-01-01). "Computational Methods for Linear Matrix Equations". SIAM Review. 58 (3): 377–441. doi:10.1137/130912839. hdl:11585/586011. ISSN 0036-1445.
  3. ^ Kitagawa, G. (1977). "An Algorithm for Solving the Matrix Equation X = F X F' + S". International Journal of Control. 25 (5): 745–753. doi:10.1080/00207177708922266.
  4. ^ Bartels, R. H.; Stewart, G. W. (1972). "Algorithm 432: Solution of the matrix equation AX + XB = C". Comm. ACM. 15 (9): 820–826. doi:10.1145/361573.361582.
  5. ^ Hamilton, J. (1994). Time Series Analysis. Princeton University Press. Equations 10.2.13 and 10.2.18. ISBN 0-691-04289-6.