Q2. What is the way to achieve stability in time delay systems?
For instance, an appropriate sliding mode strategy can achieve stabilization by “dominating” nonlinear terms and additive disturbances, provided some appropriate “matching conditions” hold (roughly speaking, the disturbances must belong to the space spanned by the input function).
Q3. What is the way to solve the problem of stability?
The results concerning robustness with respect to external disturbances rely on either H∞ design (see [20,26] and references therein), also leading to Riccati equations and LMIs, or structural approaches, such as disturbance decoupling using models over rings (see [7,8,29] and references therein).
Q4. What is the main problem of the paper?
The present paper considers uncertain systems with single / multiple, constant or time-varying state-delays and additive perturbations which may be nonvanishing.
Q5. What are the main problems of the sliding mode approach?
Time delays commonly occur in many dynamical systems and are a source of instability and poor performances even in a linear model [28].
Q6. What is the purpose of the sliding mode approach?
It aims at designing the sliding surface in such a way that the calculable set of admissible delays is maximized where “admissible” means here those that don’t destabilize the closed-loop, relay-delay system.
Q7. How can the authors determine that the system is asymptotically stable?
By using semi-definite programming, the authors find that the system (6.1) with control (4.1) is asymptotically stable for all constant delays h < 0.99.
Q8. How do the authors determine that the system is negative-definite?
By post- and pre-multiplying by S, and fixing S = (B̃T XB̃), N < 0 becomes:SZT +ZS+hα−1Âd11Â11SÂ T 11Â T d11+qh(α+β)S+hβ −1Â2d11SÂ 2T d11 < 0. (5.14)Let us assume that (5.4) and (5.5) hold.
Q9. What is the main problem of the sliding mode approach?
the combination of delay phenomenon with relay actuators makes the situation much more complex (see for instance [12] and the survey paper [31]) : designing a sliding controller without taking delays into account may lead to unstable or chaotic behaviors or, at least, results in highly chattering behaviors [9,12].
Q10. What is the contribution of this paper?
The contribution of this paper lays on the following, original points:- Concerning the design of linear sliding surface, the extension of the method [6] to the case of a delay-dependent stabilization (then, conditions are less conservative).
Q11. What is the simplest way to prove that the system is asymptotically stable?
Varying delay: Now, if the authors don’t restrict to constant delays, then according to Theorem 5, the system is asymptotically stabilized by the control law (5.1) for h(t) < 0.51 which is of course more constraining.
Q12. What is the proof of the surface s(x)?
Theorem 2 Under assumptions (A1)− (A2)− (A3), the control (4.1) makes the surface s(x) = 0 stable and globally attractive in finite time.
Q13. What is the simplest way to choose a sliding surface?
Let us choose the following sliding surface, based on some matrix X to be chosen later (via some optimization):s(x) = Sx = BT X−1x = 0,where S ∈ R(m)×n.
Q14. What is the simplest way to solve a convex optimization problem?
The reduced system is proven to be asymptotically stable for any value of the delay less than a bound which is obtained by solving a convex optimization problem.