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Chapter 2. Stability
The left hand-side of the Lyapunov equation (2.47) can written as
� �
λ − 1
(A − I)
λ + 1
−∗ (A + I) ∗ PV (A + I)(A − I) −1 − PV
= 1
λ + 1 (A − I)−∗ ((λ − 1)(A + I) ∗ PV (A + I)
− (λ + 1)(A − I) ∗ PV (A − I))(A − I) −1
= 2
λ + 1 (A − I)−∗ (−A ∗ PV A + λA ∗ PV + λPV A − PV )(A − I) −1 .
Thus the Lyapunov equation (2.47) can be equivalently written as
−A ∗ PV A + λA ∗ PV + λPV A − PV = −
(2.52)
λ + 1
(A − I)
2
∗ (A − I). (2.53)
Next we replace PV in the left hand-side of this equation by P1 +P2 +λI −I,
and we obtain
−A ∗ (P1+P2 + λI − I)A + λA ∗ (P1 + P2 + λI − I)+
λ(P1 + P2 + λI − I)A − (P1 + P2 + λ − I)
= − (λI − A) ∗ (λI − A) − (λA − I) ∗ (λA − I)
+ (λ − 1)(−A ∗ A + λA ∗ + λA − I)
= − λ 2 I + λA ∗ + λA − A ∗ A − λ 2 A ∗ A + λA ∗ + λA − I+
λ(−A ∗ A + λA ∗ + λA − I) + A ∗ A − λA ∗ − λA + I
=(−λ 2 − λ) (A ∗ A − A ∗ − A + I) ,
where we have used (2.50) and (2.51). Thus if we choose PV = 1
2λ (P1 +
P2 + λI − I), then the left hand-side of (2.53) becomes
1
2λ (−λ2 − λ) (A ∗ A − A ∗ − A + I) = −
λ + 1
(A − I)
2
∗ (A − I). (2.54)
Since this equals the right hand-side of (2.53) we see that 1
2λ (P1+P2+λI−I)
is a solution of the Lyapunov equation (2.47). �
This ends Chapter 2. In this chapter we summarized stability results for
continuous and discrete time systems.
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