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Consider a system similar to system (4.9) but without disturbances: ˙˜

x(t) =Ax˜(t) + ˜B0+Bu(t), x˜(0) =x(0) (4.10)

in which the vector_B˜_{0 can be different from B0. This system is called a nominal system} of the system (4.9). Note that the control signal u(_·) is the same for both system. It then follows from Corollary 4.2 that, sincex(0)₋x˜(0) = 0, we can bound the distance between

x(t) and x˜(t) at all time t _≥0 by _kx(t)₋x˜(t)_{kM ≤} α. The matrix M and the constant λ >0 can be found by solving the GEVP in Equations (4.2a) and (4.2b), while the radiusα is calculated by solving the quadratic program (QP)

α= 1 λmaxd∈D (B₀−B˜₀) +W d M. (4.11)

We say that a trajectoryx(·) isrobustly safe withrobustness radius α≥0if after some finite time,x(t) not only is insideSafe but also keeps a distance of at least α from the exterior of

Safe, in the metric induced by the M-norm. Mathematically,x(_·) satisfies

∃τ _≥0 : _∀t_≥τ,BM(x(t), α)⊆Safe

where BM(x, α) :={z : kz−xkM ≤α} denotes the ball with centerx and radius α. It is obvious that if the control signal u(·) is such that the trajectoryx˜(·) is robustly safe with radiusα then the trajectoryx(·)is safe because it is alwaysα-close tox˜(·). This is illustrated in Figure 4.1 on the following page, in which the dash-dotted line represents the trajectory

˜ x(t) x(t) Safe CM(Safe, α) x(0) BM(˜x(t), α) α

Figure 4.1: Illustration of robust periodic scheduling and the bound between trajectories of the nominal system and the actual system. The gray-filled rectangle isCM(Safe, α)– theα-contraction ofSafe(the outer rectangle). A tube of radiusα(dashed lines) around the nominal trajectoryx˜(·)(dash-dotted line) encloses all possible trajectories of the actual system (solid line). If ˜x(t)∈ CM(Safe, α)at anyt then

x(t)∈Safe. Hence ifx˜(·)is safe with respect toCM(Safe, α)then x(·)is safe with respect toSafe.

˜

x(_·) of the nominal system and the solid line represents the trajectory x(_·) of the actual system. A tube of radiusα (dashed lines) around the nominal trajectory encloses all possible trajectories of the actual system, i.e., at any timet, x(t) always stays within distanceα from

˜

x(t). Therefore, ifx˜(t) is at leastα away from the exterior ofSafe,x(t)must be insideSafe.

To characterize schedules u(·) that render x˜(·) robustly safe, we need the notion of set contraction by a given radius.

Definition 4.1 (α-Contraction of a Set) Given a radius α_≥0 and a subset S of a set X which is equipped with a metricd, the α-contraction of S is defined as _Cd(S, α) :=_{x_∈

X : Bd(x, α)⊆S}. 2

The α-contraction of a setS can be computed as the Pontryagin difference2 _{ofS and the ball}

Bd(0, α): Cd(S, α) =S Bd(0, α). Employing this definition, a trajectory x˜(·)is robustly safe with respect to Safe and with radiusα if and only if it is safe with respect to the set CM(Safe, α). In Figure 4.1, CM(Safe, α) is depicted by the smaller gray-filled rectangle insideSafe. Suppose that_CM(Safe, α) is not empty and system (4.10) is schedulable with respect to the safe setCM(Safe, α) and under the peak constraintk. Then a safe schedule u(_·) for system (4.10) is also safe for system (4.9) regardless of the disturbances, i.e., the

schedule is robustly safe for system (4.9). In particular, the synthesis method presented in

Section 3.3 can be used to derive a robustly safe periodic schedule u(_·).

Choosing the Affine Vector

In the above development, we have not mentioned why the vector_B˜_{0 of system (4.10) can} be different from B0 and its role in the synthesis. From Equation (4.11), the robustness

radius α depends on B˜0. Intuitively, the smaller α is the better. Thus, it is desirable to

choose _B˜_{0 so as to minimize α. We observe that for anyB}˜_{0, αλ is the maximal distance} from all points in the set W_D to the point ( ˜B0 −B0). Therefore, to minimize α, we can

select _B˜_{0 =wc+B0 wherewcis the center of the smallest ball BM(wc, r) that contains the} setW_D. Usually_Dis symmetric about a pointdc, in which case wc=W dc. For example, if d:={d∈Rq : kdk ≤β} for someβ >0 thendc=0and hence B˜0=B0.

In addition to minimizing α, we must choose B˜0 so that the system (4.10) is schedulable

with respect to CM(Safe, α) and the given peak constraint k. That is the schedulability condition in Theorem 2.6 on page 39 must be satisfied. If_B˜_{0 =wc+B0 does not satisfy this} condition, we can find a different_B˜_{0 by solving the following optimization:}

minimize ˜ B0,α,η α subject to η ∈[0,1]m, m X i=1 ηi≤k −A−1B˜0+Bη ∈ CM(Safe, α) α= 1 λmaxd∈D (B₀−B˜₀) +W d M

The objective is to minimize α while ensuring that the system (4.10) is schedulable. Solving this optimization is difficult because of the set contraction in the second-to-last constraint. Furthermore, it is a bilevel optimization problem due to the last constraint. In the following, an iterative algorithm is proposed to search for _B˜_0.

The basic idea of the algorithm is to divide the optimization problem into two sub-problems: one is to choose _B˜_{0 and calculate α, the other is to solve for the schedulability condition.} Specifically, the following steps are performed:

1. Initially select _B˜_{0 =wc+B0 and calculateα.}

2. Compute CM(Safe, α). If it is empty then report “No solution” and abort. If −A−1_B˜_0+Bη

∈ CM(Safe, α) then return B˜0 and terminate.

3. Find a new _B˜0

0 that is closest to B˜0 while satisfying the schedulability condition with

respect toCM(Safe, α); that is to solve the optimization

minimize ˜ B0 0,η B˜ 0 0−B˜0 M subject to η_∈[0,1]m, m X i=1 ηi ≤k −A−1B˜₀0 +Bη_{∈ C}M(Safe, α)

4. Calculate the robustness radius α0 corresponding toB˜0₀. If α0 ≤α then returnB˜0₀ and terminate3_.

5. If a given maximal number of iterations has been reached then terminate; otherwise, set_B˜_{0 to B}˜0

0 and α to α0, then repeat from step 2. The Robust Periodic Scheduling Synthesis Algorithm

Putting everything together, the robust periodic scheduling synthesis algorithm consists of two steps:

1. Compute_B˜_{0,α andCM(Safe, α). IfCM(Safe, α) =∅then terminate.}

2. Call Algorithm 3.4 to construct a periodic schedule u(·) for system (4.10) with the

3_{Note that ifα}0_{≤αthenCM(Safe, α)⊆ CM(Safe, α}0_{), hence−A}−1_B˜0

0+Bη

contracted safe set CM(Safe, α).

The obtained scheduleu(_·) will be robustly safe for the original system (4.9).