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Upper Bounds

Suppose that the linear time-invariant (LTI) system

˙x(t) = Ax(t) + Bu(t) + Dd(t), (7.1)

is controlled by

u(t) = F x(t), (7.2)

where x(t) ∈ Rn_{, u(t) ∈ R}m_{, d(t) ∈ R}p_{, and F denote the state vector, control input,}

Given a triplet Σ = (A, B, F ) wherein F is an arbitrary stabilizing state feedback controller, the non-fragility is defined as follows:

ρ(Σ) := sup{r|A + B(F + ∆) stays Hurwitz ∀∆ ∈ Sr}, (7.3)

wherein

Sr := {X ∈ Rm×n|r > 0, kXkmax < r}. (7.4)

It is noteworthy that Sρ(Σ) denotes the largest stabilizing hypercube with side length

2ρ(Σ), i.e., if we add any point ∆ of Sρ(Σ) to F , it will give us a stabilizing state

feedback controller F + ∆. In the sequel, we present Theorem 31 to enlighten the point that our introduced non-fragility is a strictly positive number, in other words, it is well-posed.

Theorem 31. Given a triplet Σ = (A, B, F ), the non-fragility ρ(Σ) defined by (7.3) is a strictly positive number.

Proof. Let us define the following scalar-valued function:

g(V ) := λmax(A + BV ) = λmax A + B n

X

i=1

eT_i ⊗ V (:, i)
.

We know that the function g(v) is a continuous function of v where

v := [V (:, 1)T . . . V (:, n)T]T.

Because, it is a composition of a series of continuous operations including taking maximum, taking real part, and calculating eigenvalues of affine expression of state feedback controller. Thus, defining the

and considering the continuity property of g, we can say that

g(f + δ) − g(f ) ≤ , (7.5)

holds for all kδk2 ≤ c() where c() is a strictly positive number which depends on

. To guarantee g(f + δ) < 0, we enforce g(f ) +  < 0 to hold, i.e., we choose  as follows:

 < −g(f ) = −λmax(A + BF ).

The side length of the largest hypercube inscribed by kδk2 ≤ c() (equivalently k∆kmax

in space of Rm×n_{) is equal to √}2c()

mn. Because, all the 2

mn_{corner points of the hypercube,}

i.e., (±√c() mn, . . . , ± c() √ mn) satisfies mn X i=1 δ_i2 ≤ c()2,

and particularly, the equality holds which implies the maximality of such a hypercube. Thus, regarding the non-fragility ρ(Σ) we deduce

ρ(Σ) ≥ √c()

mn, ∀ ∈ 0, −λmax(A + BF )
. (7.6)

Since √c()

mn > 0 holds, then non-fragility ρ(Σ) is strictly positive and proof is done.

Remark 16. To obtain the best lower bound on non-fragility ρ(Σ), we can take supre- mum from lower bound √c()

mn over all choices  ∈ 0, −λmax(A + BF )
. It can easily

be checked that c() is an increasing function of . Thus, the best lower bound would be lim→−λmax(A+BF )c().

Remark 17. It is noteworthy that the result of Theorem 31 may seem trivial because of r > 0 in definition of Sr. However, it is not the case and the result of Theorem 31

is nontrivial. Because, Theorem 31 basically shows that in (7.3), the set on which the supremum is taken, i.e., the following set:

{r|A + B(F + ∆) stays Hurwitz ∀∆ ∈ Sr},

is not empty. If such a fact is not shown, then taking the supremum will not be possible and consequently, the strict positivity of the non-fragility ρ(Σ) defined by (7.3) cannot be implied trivially.

Remark 18. In the rest of the chapter, for the sake of the simplicity in our notations, we drop the argument Σ from ρ(Σ) and simply use ρ.

To compute ρ in an exact way, the infinite set of feasibility problems should be considered. However, such an approach is not computationally cheap nor practical. In [125], it is mentioned that calculating the exact minimum destabilizing real pertur- bation is unfortunately impossible. Thus, we present the following theorems which suggest an analytic upper bound on ρ.

Theorem 32. Given a triplet Σ = (A, B, F ), the non-fragility ρ(Σ) defined by (7.3) is upper bounded by

ˆ

ρ = −Tr(A + BF ) kBk1

. (7.7)

Proof. The expression Tr(A + BF + B∆) is sum of eigenvalues of A + BF + B∆ and must be negative. Thus, we have

Tr(B∆) < −Tr(A + BF ), (7.8)

for all ∆ in Sρ. Since

then (7.8) holds for

∆ = (ρ − ζ)sign(BT),

In fact, the left hand side of (7.8) takes its maximum value for ∆ = (ρ−ζ)sign(BT_)
.

Equivalently, we have

Tr(B∆) = Tr B(ρ − ζ)sign(BT)
= (ρ − ζ)kBk1 < −Tr(A + BF ).

Taking the supremum from both sides, (7.7) is resulted.

Stating the following theorem, we improve the upper bound ˆρ.

Theorem 33. Given a triplet Σ = (A, B, F ), the non-fragility ρ(Σ) defined by (7.3) is upper bounded by γ ˆρ where

γ = sup{α|λmax A + BF + β ˆρBsign(BT)
< 0, ∀β ∈ [0, α]}. (7.9)

and the parameter γ is less than or equal to 1. Proof. Substituting the α = 0, we observe that

λmax A + BF + β ˆρBsign(BT)
= λmax(A + BF ) < 0.

Because, A + BF is Hurwitz. Since λmax A + BF + β ˆρBsign(BT)
is a continuous

function of β and it takes the negative value of λmax(A + BF ) at β = 0, then α > 0.

Since for γ + θ, there exists a β ∈ [γ, γ + θ] for which we have

λmax A + BF + β ˆρBsign(BT)
≥ 0,

first part of the theorem is done.

To prove the second part, for β = 1, we claim that

λmax A + BF + β ˆρBsign(BT)
≥ 0. Because, n X i=1 λi A + BF + ˆρBsign(BT)
= Tr A + BF + ˆρBsign(BT)
=

Tr(A + BF ) + Tr ˆρBsign(BT)
= Tr(A + BF ) + ˆρTr Bsign(BT)
= 0,

and since the sum of all real parts of eigenvalues is 0, then λmax cannot be less than

0. Meanwhile, the last line of the above-mentioned lines is resulted from (7.7). Thus, α cannot be greater than or equal to 1. Since α < 1, then according to the definition of supremum, it is resulted that γ ≤ 1.