Q2. What is the significant feature of a network-based feedback system?
In an NCS, the most significant feature is the network induced delays, which are usually caused by limited bits rate of the communication channels, by a node waiting to send out a packet via a busy channel, or by signal processing and propagation.
Q3. What is the space of square-integrable vector functions over [0, ]?
The space of square-integrable vector functions over [0, ∞) is denoted by L2[0, ∞), and for w ={w(t)} ∈ L2[0, ∞), its norm is denoted by ‖w‖2.
Q4. What is the main reason why time delay systems are an active research area?
Time-delay systems, also called systems with after effect or dead time, hereditary systems, equations with deviating argument or differential–difference equations, have been an active research area for the last few decades.
Q5. What is the main reason for the introduction of this new model?
Theintroduction of this new model is motivated by the observation that sometimes in practical situations, signals transmitted from one point to another may experience a few network segments, which can possibly induce successive delays with different properties due to variable network transmission conditions, and has been clearly justified by a state-feedback remote control problem.
Q6. How can the authors find the disturbance attention level bound with admissible controllers?
H∞ disturbance attention level bound with admissible controllers can be readily found by solving the following convex optimization problem:Minimize subject to (60) over P̄ > 0, Q̄ 0, R̄ 0, Z̄i > 0, i = 1, 2, M̄ > 0, K̄ , S̄, T̄ , Ū ,V̄ , and diagonal matrix W̄ > 0.
Q7. What is the minimum guaranteed closed-loop H performance?
When the authors do not consider the lower bound of the network induced delays, that is, m = 0, by using Theorem 3 (assuming that m is sufficiently small), the minimum guaranteed closed-loop H∞ performance obtained is min =3.1207.
Q8. What is the corollary of the NCS in Fig. 1?
From Theorem 3, the authors know that there exists a statefeedback controller in the form of (31) such that the closed-loop NCS in (36) is asymptotically stable with an H∞ disturbance attention level if there exist matrices P > 0, Q 0, R 0, Zi > 0, i = 1, 2, M > 0, K, S, T, U, V, and a diagonal matrix W > 0 satisfying (47).
Q9. What is the minimum guaranteed closed-loop H performance for different values of lower delay bound?
It is assumed that the network induced delays k satisfy m k M, the maximum number of data packet dropouts is 2, and the sampling period is 10 ms.
Q10. What is the common model used to represent time-delay systems?
The most commonly and frequently used state-space model to represent time-delay systems isẋ(t) = Ax(t) + Adx(t − d(t)), (1) where d(t) is a time delay in the state x(t), which is often assumed to be either constant, or time-varying satisfying certain conditions, e.g.,0 d(t) d̄ < ∞, ḋ(t) < ∞. (2) Almost all the reported results on time-delay systems are based on this basic mathematical model.
Q11. What is the condition for the existence of a desired controller?
Theorem 5. Consider the NCS in Fig. 1. Given a positive constant , there exists a state-feedback controller in the form of (31) such that the closed-loop system in (36) is asymptotically stable with an H∞ disturbance attention level if there exist matrices P > 0, P̄ > 0, W > 0, Q̄ 0, R̄ 0, X > 0, X̄ > 0, Y > 0, Ȳ > 0, Ni > 0, N̄i > 0, Zi > 0, Z̄i > 0, i = 1, 2, M̄ > 0, K̄ , S̄, T̄ , Ū , V̄ , and a diagonal matrix W̄ > 0 satisfying (61) and[−X P∗ −M ] 0, [−Y P ∗ −W ]0,[−Ni P ∗ −Zi ] 0, (63)P̄ P = I, X̄X = I, Ȳ Y = I, W̄W = The author, Z̄iZi = I, N̄iNi = I, i = 1, 2. (64) Moreover, if the above condition is feasible, a desired controller gain matrix in the form of (31) is given by (58).
Q12. What is the simplest way to quantize a sampler?
In Fig. 1, it is assumed that the sampler is clock-driven, while the quantizer, controller and zero-order hold (ZOH) are event-driven.
Q13. What is the sampling period of the sampler?
The sampling period is assumed to be h where h is a positive real constant and the authors denote the sampling instant of the sampler as sk , k = 1, . . . ,∞.
Q14. What is the problem of the closed-loop NCS in Fig. 3?
More specifically, assuming that the matrices A, B, C, D, E, F in (27) and the controller gain matrix K in (31) are known, the authors shall study the conditions under which the closed-loop NCS in (36) is asymptotically stable with an H∞ disturbance attention level .
Q15. What is the purpose of this paper?
Following the work of Lam et al. (2007), it is their intention in this paper to present new stability and H∞ performance conditions for systems with multiple successive delay components, and apply this new model to network-based control.
Q16. Why is the ZOH updated between sampling instants?
The reason is that the signal transmission delays may not necessarily be integer multiples of the sampling period, and thus the ZOH may be updated between sampling instants.
Q17. What is the simplest way to quantize a state variable?
In addition, it is assumed that the state variable x(t) is measurable, and the measurements of x(t) are firstly quantized via a quantizer, and then transmitted with a single packet.
Q18. What is the proof of the assumption that yT(t)y(t) is?
Since (23) guarantees 1 + 2 + T2 + 3 + 6 + 4 + 5 < 0, the authors have yT(t)y(t) − 2wT(t)w(t) + V̇ (t) < 0 (26)for all nonzero w ∈ L2[0, ∞).