Entrevista semi-estructurada cualitativa

Having discussed attracting sets of control systems in the previous section, we now return to the Green Scheduling problem of system (2.19) and apply this analysis tool to derive sufficient schedulability conditions. Essentially, if a subset of Safe is attracting then the

system is schedulable because, according to Definition 2.5, there exists an admissible feedback control law that drives the system’s state to this set, hence to Safe, in finite time. From

Theorem 2.9, to characterize this attracting subset, we need to find a function V :Rn→R+ of state and a number α that satisfy the conditions of the Theorem.

It is well-known in linear system theory that a Lyapunov function for a linear time-invariant system is quadratic of the formV(x) =xT_{M x, whereM ∈}

Rn×n is a positive semidefinite matrix (Rugh, 1996). Because function V in Theorem 2.9 must satisfy the Lyapunov-like differential inequality (2.21), it is natural to seek a quadratic functionV for system (2.19). A sublevel set of a quadratic function is an ellipsoid in the Rn vector space, thus we are searching for an attracting ellipsoidal subset ofSafe (if one exists). Because Safedoes not

necessarily contain the origin, this ellipsoidal subset might not center at0. Letxc∈int(Safe) be the center of this ellipsoid. Then the quadratic function V is of the form

V(x) := (x₋xc)TM(x−xc)

in which M _∈Rn×n is positive semidefinite, denotedM 0. The attracting ellipsoid we are looking for is thus given byA:=

inequality (2.21) reads, for allx_∈Rn such thatV(x)≥α2, inf u∈Usupd∈D 2 (x−xc)T M(Ax+B0+Bu+W d) =2 (x₋xc)T M A(x−xc) + 2 inf u∈Usupd∈D (x₋xc)T M(Axc+B0+Bu+W d) ≤ −γ. (2.22)

As with Lyapunov functions, M is not any positive semidefinite matrix but must also satisfy the Lyapunov inequality with respect to the dynamics (2.19) (Rugh, 1996): AT_{M+M A0.} Moreover, it is desirable to obtain the maximum decay rate of V so that we can bound V(x(t))as in Equation (2.21). To this end, we consider the generalized eigenvalue problem (GEVP)

maximize

λ,M λ

subject to M 0

ATM+M A −2λM

in whichλ, being the decay rate ofV(x(t)), is maximized. Suppose that the above GEVP has an optimal solution with λ >0. We then have

2 (x−xc)TM A(x−xc)≤ −2λ(x−xc)TM(x−xc) ∀x∈Rn.

It follows that if for all x_∈Rn such thatV(x)≥α2,

−2λ(x−xc)T M(x−xc) + 2 inf

u∈Usupd∈D

(x−xc)T M(Axc+B0+Bu+W d)≤ −γ (2.23)

then the inequality (2.22) is verified. In particular, if the static game in Equation (2.23) (i.e., the inf-sup term on the left-hand side) is non-positive for all x then the inequality is satisfied. We observe that in Equation (2.23), the optimization variables u and d are

decoupled, therefore inf u∈Usupd∈D (x−xc)T M(Axc+B0+Bu+W d) = (x₋xc)TM(Axc+B0) + inf u∈U(x−xc) T _{M Bu+ sup} d∈D (x₋xc)TM W d = sup d∈D inf u∈U(x−xc) T _M(Ax c+B0+Bu+W d).

BecauseU is finite, the inner infimum is equivalent to the minimization

min

u∈U (x−xc)

T _M(Ax

c+B0+Bu+W d).

Letco(_U)denote the convex hull of_U in the continuous spaceRm. Then this minimization has the same optimal value as min_u∈co(U)(x₋xc)T M(Axc+B0+Bu+W d). Indeed, assume

otherwise then there existsu? _∈co(_U) such that

(x−xc)T M(Axc+B0+Bu?+W d)<min

u∈U (x−xc)

T

M(Axc+B0+Bu+W d).

From the definition ofco(_U) (Boyd and Vandenberghe, 2006, sec. 2.1.4),u? =θ1u(1)+· · ·+

θku(k) whereu(i) ∈ U and θi≥0for i= 1, . . . , k, and θ1+· · ·+θk= 1. It follows that

(x−xc)TM(Axc+B0+Bu?+W d) = k X i=1 θi(x−xc)T MAxc+B0+Bu(i)+W d ≥ k X i=1 θimin u∈U(x−xc) T _M(Ax c+B0+Bu+W d) = min u∈U (x−xc) T _M(Ax c+B0+Bu+W d),

a contradiction. Therefore Equation (2.23) is equivalent to

−2λ(x₋xc)TM(x−xc) + 2 sup d∈D min u∈co(U)(x−xc) T_M(Ax c+B0+Bu+W d)≤ −γ. (2.24) We observe that sup_d∈Dmin_u∈co(U)(x₋xc)TM(Axc+B0+Bu+W d) ≤ 0 ∀x ∈ Rn im-

plies the inequality (2.22) being satisfied. This leads to the following result on attracting sets of control system (2.19). Its proof is given in Appendix A.1.10 on page 186.

Lemma 2.6 Suppose there exist M _∈Rn×n andλ >0 such that

M 0, (2.25a)

ATM +M A −2λM. (2.25b)

Furthermore, there exists xc∈Rn satisfying for alld∈ D, there exists u∈co(U) such that Axc+B0+Bu+W d= 0. Then for anyα >0,A:={x∈Rn : (x−xc)TM(x−xc)≤α2} is an attracting set of control system (2.19)with basin of attraction _B:=Rn. 2

The validity of Lemma 2.6 depends on the existence of M and λ. Recall that the state matrix A is assumed to be Hurwitz. Similar to the existence of Lyapunov functions for asymptotically stable linear systems (Rugh, 1996), there always existM andλsatisfying the hypothesis of Lemma 2.6. This result is confirmed in Proposition 2.1, whose proof can be found in Appendix A.1.11 on page 186.

Proposition 2.1 If the state matrix A is Hurwitz then there exist M _∈ Rn×n and λ >0

that satisfy the conditions in Lemma 2.6. ₂

As mentioned at the beginning of this section, if Safe contains an attracting subset then

the system is schedulable. This is illustrated on the phase space of control system (2.19) in Figure 2.7 on the next page. The gray-filled ellipsoid in this figure is the attracting subset

A :={x ∈ Rn : (x−xc)T M(x−xc) ≤ α2} of Safe. Outside this subset, there exists a control inputu∈ U such thatx(t)is always attracted towardsA(indicated by the arrows) for

all disturbancesd_{∈ D}, thus it is safe. Therefore the system is schedulable. Using Lemma 2.6, the following sufficient schedulability condition is straightforward, hence its proof is omitted. Proposition 2.2 If there exists xc∈int(Safe) such that

(x−xc)TM(x−xc)≤α2

xc

Safe

Figure 2.7: Illustration of an attracting subset ofSafefor sufficient schedulability condition: the gray-filled subset{x∈Rn : (x−xc)TM(x−xc)≤α2} ofSafeis attracting. Outside this subset, there exists a controlu∈ U that drivesx(t)towards the subset (indicated by the arrows) for all disturbancesd∈ D, hence making it safe.

then the system (2.19) is schedulable. ₂

Proposition 2.2 requires us to explicitly find a pointxc∈int(Safe)that satisfies its condition. However, for determining whether a system is schedulable, it is sufficient to verify the existence of such a point. The sufficient schedulability condition stated in Theorem 2.10 below can be checked by performing standard geometric operations on the setsSafe, Dand

U. First, let us introduce several geometric operations on sets.

• The negation1 _{of a set X⊆}

Rn

−X:=_{−x_∈Rn : x∈X}.

• The sum of a setX _⊆Rn and a vectorv∈Rn

X+v:={x+v∈Rn : x∈X}.

1_{Do not confuse this with set differenceX\Y}

:={x∈X : x6∈Y}and set complementXC_:=

• The product of a matrixA_∈Rm×nand a set X⊆Rnis a set in Rm

AX :={Ax∈Rm : x∈X}.

• The Pontryagin difference of two setsX, Y _⊆Rn

X Y :={z∈Rn : z+y∈X∀y∈Y}={z∈Rn : Y +z⊆X}.

• The Minkowski sum of two setsX, Y _⊂Rn

X_⊕Y :={x+y_∈Rn : x∈X, y ∈Y}.

Using these set operations, the condition in Proposition 2.2 is equivalent to

∃xc∈int(Safe) : Axc+B0+WD ⊆ −Bco(U). (2.26)

We can now state the main sufficient schedulability condition for system (2.19).

Theorem 2.10 Ifint(Safe)∩Q6=∅, whereQ:=−A−1Bco(U) A−1(B0+WD), then the

system (2.19) is schedulable. ₂

Proof See Appendix A.1.12 on page 186.

Checking the schedulability condition in Theorem 2.10 does not require solving the GEVP (Equations (2.25a) and (2.25b)) to compute the matrixM and the value λ. In practice, the setsco(_U),Safe and_Dare usually hyper-rectangles or polytopes. Therefore, checking the

schedulability condition in Theorem 2.10 involves only standard operations on polytopic sets, which can be computed numerically with readily available scientific software, for instance the Multi-Parametric Toolbox for MATLAB™ (Kvasnica et al., 2004) and the Parma Polyhedra Library (BUGSENG, 2012). For the n-choose-k case, i.e., U = {u∈ {0,1}m _{: kuk}

1 ≤k}

{u_∈[0,1]m _{: kuk}

1 ≤k} – a polytope.

Remark 2.4 A minor catch in Theorem 2.10 is that the interior operator (int(Safe)) might not be supported in some scientific computation software. In that case, we can replace

(int(Safe)∩Q)with(Safe∩Q). It is possible to show thatint(Safe∩Q)⊆int(Safe)∩Q.2