La propuesta y el diseño de la investigación

Using the same approach as in Section 2.3.1, we investigate periodic control signals for system (2.9) to derive a sufficient schedulability condition for it.

Recall that aδ-periodic control signalu(·), whereδ >0, satisfiesu(t+δ) =u(t)∀t≥0. The utilizationηi of periodic control inputiis the fraction of time in a period whereui= 1, and is defined asηi := 1_δR₀δui(t)dt∈[0,1]. The utilization vectorη∈[0,1]m of all control inputs isη := [η1, . . . , ηm]T.

A standard tool to analyze a linear system under periodic control signals is the averaging technique (see Tokarzewski, 1987; Sun and Ge, 2005). Given aδ-periodic control signalu(·) with utilization vector η, the average system of (2.9) with respect to η is defined as the time-invariant affine dynamical system

˙

x(t) =Ax(t) + (B0+Bη) (2.10)

with state variable x and starting from the same initial state: x(0) =x(0). Note that the average system is autonomous, meaning that it does not have any control input. For each initial statex(0)and a fixed utilization vector, the trajectory x(_·) of the average system is unique. Since A is Hurwitz, the average system is uniformly exponentially stable and its trajectories always converge to the unique equilibrium x? =₋A−1(B0+Bη)(Rugh, 1996).

Note that A is invertible because it is Hurwitz. If x? is in the interior of Safe, denoted

int(Safe), then obviously x(·) will be safe. If, in addition, the trajectoryx(·)of system (2.9) always stays close enough to x(·) regardless of the initial state thenx(·) is also safe and thereforeu(·) is a safe schedule. More precisely, if there exists >0 such that for any initial state x(0) =x(0)∈ X0,

2. B(x?_{, )⊆Safe}

then x(_·) is safe. Here, the symbol _k·k denotes both the Euclidean vector norm and the corresponding induced matrix norm. The notationB(c, r)denotes the ball with centercand radiusr, i.e.,B(c, r) :={x : kx−ck ≤r}. Indeed, we will study the state error(x(t)−x(t)) and will show that for any >0, there existsδ >0and a δ-periodic control signalu(·) such thatx(t) is always-close tox(t)for all t≥0.

State Error

Let ξ(t) = x(t) ₋x(t) _∀t _≥ 0 be the state error between system (2.9) and its average system (2.10). Intuitively, as the period δ gets smaller, the trajectory x(_·) gets closer to the trajectoryx(_·), that islimδ→0kx(t)−x(t)k= limδ→0kξ(t)k= 0 for allt≥0. In what

follows, we justify this intuition by directly calculating the state error ξ(t) and deriving an upper-bound on its norm.

From the differential equations of the dynamical systems we obtain the dynamics of the error: ˙

ξ(t) =Aξ(t) +B(u(t)₋η), ξ(0) = 0. (2.11)

Note that the errorξ(t)does not depend on the initial statex(0). Its solution can be written explicitly as (Rugh, 1996)

ξ(t) = Z t

0

eA(t−s)B(u(s)−η)ds, t≥0. (2.12)

Letσ =_bδt_c, where_bc_c denotes the largest integer not exceeding c. Then we have ξ(t) = σ−1 X i=0 Z (i+1)δ iδ eA(t−s)B(u(s)−η) ds+ Z t σδ eA(t−s)B(u(s)−η)ds (2.13)

as Z (i+1)δ iδ eA(t−s)B(u(s)−η) ds= eA(t−(i+1)δ Z (i+1)δ iδ eA((i+1)δ−s)B(u(s)−η) ds

and use the fact thatu(_·) is δ-periodic to obtain

= eA(t−(i+1)δ) Z δ 0 eA(δ−s)B(u(s)−η) ds = eA(t−(i+1)δ)ξδ in whichξδ =ξ(δ) = Rδ

0 eA(δ−s)B(u(s)−η) dsis the state error after one periodδ. Similarly,

the tail integral in Equation (2.13) can be written as Z t σδ eA(t−s)B(u(s)₋η) ds= Z t−σδ 0 eA((t−σδ)−s)B(u(s)₋η) ds=ξ(t₋σδ)

whereξ(t₋σδ) is the state error at time instant (t₋σδ). Therefore,

ξ(t) = σ−1 X i=0 eA(t−(i+1)δ) ! ξδ+ξ(t−σδ). (2.14)

The following Lemma gives us a uniform upper-bound on _kξ(t)k, independent of time t. Its proof can be found in Appendix A.1.5 on page 172.

Lemma 2.3 There exist finite and positive constants α,β and γ, which are independent of the time period δ and the control signal u(_·), such that

kξ(t)k ≤ 1

2kAkγβ

2 δ2

1₋e−αδ +γβδ, ∀t≥0. 2

It is worth noting that in Lemma 2.3 the constantsα andβ depend only on the dynamics (2.9), in particular the matricesA andB, and the constantγ depends only onη. In addition,

the obtained upper-bound onkξ(t)_k holds for all timet.

point-wise distance betweenx(_·) andx(_·). Obviously_kξ(_·)_k∞ is bounded above by the same upper-bound in Lemma 2.3. By simple calculations and the l’Hôpital’s rule, we can show that this upper-bound goes to 0as δ_→0. The following Lemma confirms the intuition that

x(·) gets closer to x(·) asδ gets smaller.

Lemma 2.4 The state error between system (2.9) under δ-periodic control signals and its average system (2.10) vanishes as δ goes to 0. That is limδ→0kξk_∞ = 0 for all δ-periodic

control signals u(_·). ₂

Lemma 2.4 implies that for any >0, there existsδ>0 such that for all 0< δ≤δ, any δ-periodic control signalu(_·) with utilization η will drive the trajectoryx(_·) to be -close to the trajectoryx(_·) of the average system, regardless of the initial statex(0).

Remark 2.2 It can be proved that Lemmas 2.3 and 2.4 still hold for control signals u(_·) that are notδ-periodic but only satisfy the utilization constraint 1_δR(i+1)δ

iδ u(s)ds=η,∀i∈N.

This enables conventional real-time scheduling algorithms (Liu, 2000) to be used to schedule the system. However, to derive schedulability conditions, we only need to consider periodic

control signals. 2

Sufficient Schedulability Condition

Now that we have shown the state trajectory of the original system and its average system can be made arbitrarily close, we only need to construct aδ-periodic control signal u(_·)such thatku(t)k1≤k∀t≥0 to complete the schedulability condition. Similarly to Theorem 2.2

on page 28, this construction is always possible if Pm

i=1ηi ≤k. Therefore, we arrive at a sufficient schedulability condition stated in the following Theorem, whose proof is given in Appendix A.1.6 on page 174.

Theorem 2.6 Given a peak constraint k on the control signal u(_·), k _{∈ {}0,1, . . . , n_}. If there exists η_∈[0,1]m _{such that}

1. _kη_k1 =Pm

i=1ηi ≤k, and

then the system (2.9) isk-schedulable. ₂

Necessary Schedulability Condition

The next Theorem provides a necessary condition for the system to be k-schedulable. Its proof can be found in Appendix A.1.7 on page 174.

Theorem 2.7 Given a peak constraint k on the control signal u(·), k∈ {0,1, . . . , n}. If the system (2.9) isk-schedulable then there exists η_∈[0,1]m _{such that}

1. _kη_k1 =Pm

i=1ηi ≤k, and

2. ₋A−1(B0+Bη)∈Safe. 2

Observe that although the necessary schedulability condition in Theorem 2.7 seems to complement the sufficient schedulability condition in Theorem 2.6, there is a small gap between them: one uses the interior of Safe while the other uses the entireSafe. However,

the difference (Safe₋int(Safe))is often negligible in practice.