According to this methodology, the unique and non-negative definite stabilizing solution of the CARE (or DARE) can be obtained by computing the stable invariant subspace (or deflating subspace) of the associated Hamiltonian matrix (or symplectic matrix pencil).
In this section, we prove that Equation (1.2) has a unique positive definite solution and construct an interval that includes that solution.
Duan et al. [6] studied the equation (X-sum_{i=1}^{m}N_{i}^X^{-1}N_{i}=I), they used the Thompson metric to prove that the matrix equation always has a unique positive definite solution and they derived a precise perturbation bound for the unique positive definite solution.
In the light of Lemma 1.5, the map (F(P)) has a unique fixed point in Ω. Combing (1) and (2) shows that (1.3) has a unique positive definite solution (P_{0}), and (widetilde{P}_{1}leq P_{0}leqwidehat {P}_{1}).
If the conditions of Theorem 3.1 are satisfied, then the sequence (P^{ k)}) given by (4.1) is convergent and converges to the unique positive definite solution (P_{0}) of the discrete algebraic Riccati equation (1.3).
According to Theorem 3.1, the discrete algebraic Riccati equation (1.3) has a unique positive definite solution (P_{0}), and (widetilde{P}_{1}leq P_{0}leqwidehat{P}_{1}). Let (e k)=|P^{ k+1)}-P^{ k)}-P^{ kthe iteration error at the kth iteration, k denote the iteration number, where we choose (varepsilon=10^{-8}) and (P^{(0)}=widetilde{P}_{1}) and (P^{(0)}=Q), respectively.
The above examples show that the recursive formula defined by (6.11) is feasible and effective to compute the unique positive definite solution of Equation (1.2).
Assume next that the described approximations are adopted instead the original local PDAFs and that the recursions (20) converge to the unique positive definite matrices (bar {P}_{i}) defined by (21).
Each of these means except t≠0 arises as a unique positive definite solution of a non-linear matrix equation, satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means.
And definite ominous orchestral undertones.
Donnellan K (1966) Reference and definite descriptions.
