Q2. What future works have the authors mentioned in the paper "Robust stability and performance analysis of 2d mixed continuous-discrete-time systems with uncertainty" ?
Several directions can be explored in future works. In particular, one could investigate the possibility of generalizing the results proposed in this paper to marginally stable systems. Another possibility could be the extension of the proposed results to systems with time-varying uncertainties. A last possibility could be the extension to the design of robust controllers.
Q3. What is the proposed LMI conditions for establishing robust exponential stability?
The proposed LMI conditions are based on the introduction of complex Lyapunov functions depending polynomially or rationally on a frequency and polynomially on the uncertainty.
Q4. What is the purpose of this paper?
This paper has proposed LMI conditions for establishing robust exponential stability and for determining the robust H∞ and H2 norms of 2D mixed continuous-discrete-time systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set.
Q5. What is the history of 2D mixed continuous-discrete-time systems?
The study of 2D mixed continuous-discrete-time systems has a long history, with some early works such as [1,2] introducing basic models, systems theory and stability properties.
Q6. What is the LMI condition for determining whether a complex matrix polynomials is?
The LMI condition provided by Theorem 2 is sufficient for any chosen degrees of V (ω, p), Ti(ω, p) and Ui(ω, p), and it is also necessary for sufficiently large degrees under the assumption that P is strongly compact.
Q7. What is the condition for solving an LMI?
Since establishing whether a complex matrix polynomials is SOS is equivalent to an LMI as explained in Section 2.2, and since X(ω, p) and Y (ω, p) are affine linear matrix functions of the decision variables ζ, V (ω, p), Ti(ω, p) and Ui(ω, p), it follows that the condition (32) amounts to solving an LMI feasibility test.
Q8. What is the simplest way to quantify the feasibility of the optimization problem?
0. (35)Consequently, one has that(1)–(3) is robustly exponentially stable m∃d : ζ∗ > 0. (36)The optimization problem (34), which amounts to minimizing a linear function subject to SOS constraints and linear equations, is a convex optimization problem, in particular a semidefinite program (SDP).
Q9. What is the LMI condition for establishing upper bounds on p?
The LMI condition provided by Theorem 3 is sufficient for any chosen degrees of V (ω, p), Ti(ω, p) and Ui(ω, p), and it is also necessary for sufficiently large degrees under the assumption that P is strongly compact.
Q10. what is the simplest way to measure the feasibility of the condition trace(V (0?
Let us observe that the condition trace(V (ω0, p0)) = 1 can be introduced without loss of generality since V (ω, p) and ζ are defined up to a positive scale factor in the LMI condition provided by Theorem 2.
Q11. what is the problem of determining whether a matrix polynomial is robustly exponential?
This matrix polynomial is said to be SOS if there exist real matrix polynomials Ji(ω, p), i = 1, . . . , l, such thatJ(ω, p) =l ∑i=1Ji(ω, p) TJi(ω, p). (12)A necessary and sufficient condition for establishing whether J(ω, p) is SOS can be obtained via an LMI feasibility test.
Q12. What is the definition of the SOS matrix polynomials?
From the definition of SOS matrix polynomials in Section 2.2, the first constraint in (32) implies thatX(ω, p) ≥ 0 ∀ω ∈ R ∀p ∈ Rq.Similarly, one obtains that Y (ω, p), Ti(ω, p) and Ui(ω, p) are positive semidefinite for all ω ∈ R for all p ∈ Rq. Next, let ω0 ∈ R and p0 ∈ P be arbitrary values.
Q13. What is the condition for robust exponential stability of the system?
The system (1)–(3) is robustly exponentially stable if and only if|λi(F1(jω, p))| < 1 ∀i = 1, . . . , nd ∀ω ∈ R ∀p ∈ P. (25)This lemma can be exploited to derive a condition for robust exponential stability of the system (1)–(3) through the use of suitable Lyapunov function candidates.
Q14. What are the key contributions of researchers to the study of 2D mixed continuous-discrete time?
Researchers have investigated several fundamental properties of 2D mixed continuous-discrete-time systems, in particular stability, for which key contributions include [3,6–9].
Q15. What is the condition for robust exponential stability of P?
I− na ∑i=1ai(p)Ui(ω, p).(33)Moreover, if P is strongly compact, this condition is not only sufficient but also necessary for (1)–(3) to be robustly exponentially stable.