Systems and Control
Broad in scope, this text shows the multidisciplinary role of dynamics and control, presents neural networks, fuzzy systems, and genetic algorithms, and provides a self-contained introduction to chaotic systems.
TABLE OF CONTENTS Systems and Control
4.5: Discrete-Time Lyapunov Equation
4.5 Discrete-Time Lyapunov Equation
We will now use the Lyapunov approach to the stability analysis of the discrete-time linear time-invariant systems modeled by
By analogy with the continuous-time case, consider a quadratic positive definite form,
We evaluate the first difference of V, defined as ? V = V ( x( k + 1)) ? V ( x( k)), on the trajectories of (4.16) to obtain
where we replaced x( k + 1) in the above with Ax( k). For the solutions of (4.16) to decrease the "energy function" V, at each k > k 0, it is necessary and sufficient that ? V ( x( k)) < 0, or, equivalently, that for some positive definite Q,
Combining (4.16) and (4.18), we obtain the so-called discrete-time Lyapunov equation,
Theorem 4.3
The matrix A has its eigenvalues in the open unit disk if and only if for any positive definite Q the solution P to (4.19) is positive definite.
| Proof | ( ?) We can use the Kronecker product, and in particular relation (A.62), to represent the discrete-time Lyapunov equation as  where ? denotes the Kronecker product and s( ) is the stacking operator. It is easy to see that the above equation has a unique solution if and only if  where { ? i} is the set of eigenvalues of A. By assumption, ? |
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