![Fig. 4. Reachable region (F1(Tf), F2(Tf)) for Tf = 10, u1 ∈ [−1, 1] , u2 ∈ [−0.6, 0.6] and starting point F1(0) = 1, F2(0) = 2.](/figures/fig-4-reachable-region-f1-tf-f2-tf-for-tf-10-u1-1-1-u2-0-6-0-3r4e43ky.webp)
![Fig. 5. Development of reachable region when the transfer time Tf is varied, for u1 ∈ [−1, 1] , u2 ∈ [−0.6, 0.6] and initial condition F1(0) = 1, F2(0) = 2. The originally desired arrival time was Tf,des = 6 sec.](/figures/fig-5-development-of-reachable-region-when-the-transfer-time-1gkxaw29.webp)
Q2. What have the authors stated for future works in "Flatness-based fault tolerant control of a nonlinear mimo system using algebraic derivative estimation" ?
Future work will focus on the fault tolerant control of uncertain nonlinear systems.
Q3. What is the first condition for the control law?
The first condition only depends on the convergence rate of the derivative estimation method and is fulfilled for slowly varying actuator faults.
Q4. What is the way to control a dynamically severe fault?
In the case of a dynamically severe fault, it is sufficient to enlarge the arrival time of the system, while keeping the same stationary values.
Q5. What is the method used in the paper?
The authors use a recently published derivative estimation method (Fliess et al., 2004; Fliess et al., 2005a; Mboup et al., 2007) 1 which is based on differential algebraic manipulations of a polynomial function of time.
Q6. What is the corresponding value of the reference trajectory?
Tf denotes the system transfer time between two stationary regimes, and Fi(0) and Fi(Tf) denote the corresponding inital and stationary value of the reference trajectory.
Q7. What is the graphical representation of reachable regions?
For the fault classification scheme based on the graphical representation of reachable regions, it is important to recall that any steplike actuator fault fa,i of amplitude Fa,i leads to a shift of −Fa,i with respect to the free control signal u′i(t) for the fault-free case.
Q8. how long can the red point be reached?
If the originally designated transfer time Tf,des was set to be 6 sec and a maximum transfer time of 15 sec is fixed then the two red points that represent two intended stationary points represent a situation with a Dynamically Severe Fault and a Severe Fault: in the first case, the red point can still be reached, though in a larger transfer time of 6 sec < Tf < 9 sec, whereas in the second case the red point will not be reachable within the maximum time of 15 sec.
Q9. How do the authors achieve fault tolerant behavior of the controllers?
Fault tolerant behavior of the controllers, in this case, is achieved by adding a negative estimate of the actuator fault on the control expressions of the fault-free case.
Q10. What is the main topic of this paper?
In the field of fault tolerant control, recently, encouraging results were obtained applying algebraic techniques within the control of a three tank system (Fliess et al., 2005b).
Q11. What is the main idea of the paper?
Among all the available literature the authors would like to single out the ideas in (White and Speyer, 1987; Massoumnia, 1989) which involve the design of detection filters that are based on a geometric setting.
Q12. What is the effect of the control saturations?
As a consequence, the authors will observe perfect tracking, that is, F1(t) = F ⋆ 1 (t), F2(t) = F ⋆ 2 (t) as long as the nominal control signals u⋆i (t), i = 1, 2, reside fully within their respective saturation intervals [−Si, Si]; in this case, u ′ i = u ⋆ i (t), i = 1, 2, is valid.
Q13. What is the graphical representation of the so-called reachable regions?
The authors now use a graphical display of the so-called reachable regions, by which the authors understand the 2-dimensional set of stationary points (F ⋆1 (Tf), F ⋆2 (Tf)) that can be chosen as the final values of the reference trajectories F ⋆1 (t), F ⋆ 2 (t), once the analytic form of the reference trajectories is fixed.