Identificar el cumplimiento de las obligaciones tributarias de los comerciantes y

As we will see later in the illustrative example in Section 4.2.4, robust periodic scheduling usually induces frequent switching of the control inputs ui. This is because in order for the system to be robustly safe, the trajectory of the nominal system (4.10) must be kept within a small subset of the safe setSafe which isα-away from the exterior ofSafe. The

distanceα is therefore called the robustness radius. We showed in the previous subsection that α is not any value but depends on the dynamics of the system and the constraint of the disturbances (Equation (4.11)). Essentially,α represents the worst-case bound of the

error between the trajectories of the original system (4.9) and the nominal system (4.10). This worst-case scenario means that normally the distance between the two trajectories at any time tis less than α, as illustrated in Figure 4.1 on page 89. This observation will be exploited to reduce the switching frequency of robust periodic scheduling.

Consider any timet0 _≥0 at which the robust periodic schedule switches from control vector

u(−)_{∈ U to control vector u}(+)_{∈ U. At timet}0 _{we havekx(t}0_)−x˜(t0_)k

M ≤α. As discussed above, normally the distance between the trajectories is less thanα. Also, we have shown in Section 4.2.1 that as long asx(t) is within the distance ofα from x˜(t), the control input derived for the nominal system will be safe for the original system. It follows that, instead of switching to control vector u(+) immediately at time t0 whenkx(t0)−x˜(t0)kM < α, we can delay the switching until the first time that kx(t)−x˜(t0)kM = α, for t≥t0. In other words, we “freeze” the time and evolution of the nominal system at t0 and x˜(t0), and let the actual system continue to evolve with input u(−) _{untilx(t) is about to leave the robustness} ball BM(˜x(t0), α); at the time we “unfreeze” the nominal system and switch to input u(+).

Since the new scheduling algorithm requires monitoring the state of the system, i.e., state feedback, to detect the event kx(t)₋x˜(t0)_kM =α, we will call it anevent-triggered feedback

˜ x(t0) x(t) BM(˜x(t), α) x(t0₎ withu(−)

(a) At timet0_{, the nominal robust peri-} odic schedule switches fromu(−)

∈ U

to u(+)

∈ U. The nominal system

(˜x(_·)) is “frozen” at timet0 _{while the} original system (x(_·)) continues.

˜ x(t0_)≡x(t˜ 00₎ x(t00₎ BM(˜x(t), α) x(t0₎ withu(−) withu(+)

(b) Whenx(t)hits the boundary ofBM(˜x(t0), α)(gray-filled) at timet00_≥t0_{, the nominal system is resumed and the control} input for both systems is switched tou(+)_.

Figure 4.2: Event-triggered feedback scheduling based on robust periodic scheduling. The trajectory˜x(_·) of the nominal system is drawn in dash-dotted lines, while the trajectoryx(_·)of the original system is drawn in solid lines.

scheduling algorithm. Figures 4.2a and 4.2b on the current page illustrate the new scheduling

algorithm. In Figure 4.2a, both trajectoriesx˜(_·)(dash-dotted line) andx(_·)(solid line) evolve with control vector u(−)_{. However, while x˜(·) is “frozen” at time instantt}0_{, x(·) continues for}

t_≥t0 as long as x(t)is inside BM(˜x(t0), α). In Figure 4.2b, whenx(t) hits the boundary of the robustness ball at time instantt00_≥t0, the evolution of the nominal system is resumed at statex˜(t00)≡x˜(t0)and the control input (for both systems) is switched tou(+). It is obvious that the nominal trajectoryx˜(·) does not change in shape but only in time. That is if we remove all the time intervals during which the nominal system is “frozen” we will recover the original nominal trajectory. Thereforex˜(·) is still robustly safe with robustness radiusα. Becausex(t) is always maintained within distanceα fromx˜(t),x(_·) must be safe.

Let us formulate mathematically the resulted schedule. Any nominal δ-periodic schedule

˜

u(_·) of the form in Equation (3.1) can be represented by an infinite sequence of pairs (u(0), τ0),(u(1), τ1), . . . that satisfy the following conditions:

• For alli= 0,1, . . .,u(i) is a valid control vector inU andτi>0is the time duration thatu(i) is applied;

• For alli= 0,1, . . .,u(i) _6=u(i+1)_;

• There exists s > 1 such that u(i) = u(i+s) and τi = τi+s for all i = 0,1, . . ., and Ps−1

i=0τi =δ.

At any time t≥0,u˜(t) is determined by

˜ u(t) =        u(0) if 0≤t < τ0 u(i) if Pi−1 j=0τj ≤t <Pji=0τj,i≥1.

With the event-triggered feedback scheduling strategy, the resulted scheduleu(_·)goes through the same sequence of control vectors as the nominal periodic scheduleu˜(_·) but with extended time durations. In particular, it is represented by a sequence (u(0), τ₀0),(u(1), τ₁0), . . . with τ_i0 ≥τi for all i = 0,1, . . . Note that u(·) is no longer periodic in time as u˜(·) but is still periodic in the sequence of control vectors (i.e.,u(i)=u(i+s) for alli).

The flowcharts in Figure 4.3 on page 96 compare the algorithms for periodic scheduling and event-triggered feedback scheduling based on robust periodic scheduling. On the far left, the basic periodic scheduling algorithm has a main loop which simply applies each control vectoru(i) _{for a fixed durationτ}

i in the correct sequence. In the middle, the event-triggered scheduling algorithm adds two new steps (gray-filled blocks) to the basic algorithm:

• The first block calculates the state of the nominal system after the delayτi, based on Equation (4.10);

• The second block monitors the state of the actual system and detects the event when

kx(t)−x˜(t0)kM =α.

We note that there is a possibility that the event-triggered scheduling algorithm will result in an unsafe scheduleu(_·). Suppose at time t0, x˜(t0)_{6∈ C}M(Safe, α) and the nominal system is “frozen.” Also, the disturbances on the system are such that x(t) never reaches the boundary of the robustness ball around x˜(t0), hence the event _kx(t)₋x˜(t0)_kM =α never

Algorithm 4.1 Event-triggered Feedback Scheduling Based on Robust Periodic Scheduling Input: the sequence(u(0), τ0),(u(1), τ1), . . . of the nominal periodic schedule, the nominal

dynamics (4.10),M,λ, maximal waiting time τmax

1: i_←0

2: x˜_←x(0)

3: whiletruedo . main loop repeats indefinitely

4: Apply u(i) 5: Delay for τi

6: x˜←eAτix˜+ eAτi−I

A−1( ˜B0+Bu(i)) . update x˜

7: t0 _←t . mark the current time

8: repeat .monitor state and detect event

9: Measure (monitor) x(t)

10: until(_kx(t)₋x˜_kM =α)_∨((˜x_{6∈ C}M(Safe, α))∧(t−t0 ≥τmax))

11: i_←i+ 1

12: end while

occurs. Consequently, the nominal system is stuck at timet0 and statex˜(t0) indefinitely and therefore the system is not safe. This situation can be avoided by restricting the waiting time for the event to a finite maximal value, denoted by τmax. A detailed algorithm of the

event-triggered scheduling strategy is listed in Algorithm 4.1 on the current page, in which the symboltdenotes the current time when the corresponding statement is executed. An illustrative example will be given in Section 4.2.4.