Q2. What is the way to verify convergence of the algorithm?
when verifying convergence of the algorithm, it is necessary to begin checking the convergence of the vectors only after the β̃k and β̂k values are nearly equal.
Q3. What is the condition for the closed-loop system?
The closed-loop system is said to be:• well-posed if det (I − P11(∞)∆1) 6= 0. This is the necessary and sufficient condition that all closed-loop transfer functions in Figure 5.1 be proper.• stable if all closed-loop transfer functions in Figure 5.1 are analytic in the closed righthalf-plane.
Q4. How many conditions can be satisfied by a matrix?
Choose α > 0 small enough so that the three conditionsI − αZ > 0 Σ22 − σ21I + αL < 0σ21 (V ∗ZV − U∗ZU)− αT (Σ22 − σ21I + αL )−1 T ∗ < 0are satisfied.
Q5. What is the simplest way to determine the stability of a given quadratic set?
The authors have shown that quadratic stability with respect to such a parameter can be ascertained by determining if the convex set{X =[P 0 0 D2] : P ∈ Cn×n, D2 ∈ Cm×m, X = X∗ > 0,M∗XM −X < 0 }is nonempty.
Q6. What is the simplest way to compute the lower bounds for a structured singular value?
The algorithm resembles a mixture of power methods for eigenvalues and singular values, which is not surprising, since the structured singular value can be viewed as a generalization of both.
Q7. What is the condition that all closed-loop transfer functions in Figure 5.1 be proper?
6= 0. This is the necessary and sufficient condition that all closed-loop transfer functions in Figure 5.1 be proper.• stable if all closed-loop transfer functions in Figure 5.1 are analytic in the closed righthalf-plane.
Q8. Why is the constant matrix feedback system called unstable?
Motivated by connections with stability of systems, which will be explored in detail in the sequel, the authors call this constant matrix feedback system “unstable”.