B

Proof of Theorem 8.2.2

We begin with a preliminary lemma.

Consider the nonlinear time-varying system:

where x(t) ∈ℜⁿ is the state of the system and w(t) ∈ℜ^r is an input, together with the cost functional:

Suppose the following assumptions hold:

(a1) ϕ(., .) is continuous with respect to both arguments;

(a2) For all {x(t), w(t)} ∈ L₂[0, ∞), the integrals (B.2) above are bounded:

(a3) For any given ∈ > 0, w ∈ W, there exists a δ > 0 such that for all x₀ ≤ δ, it implies:

$\left|J_i\left(x_1\left(t\right),w\left(t\right)\right)-J_i\left(x_2\left(t\right),w\left(t\right)\right)\right|<\in ,i=0,\mathrm{\dots },l,$

where x₁(.), x₂(.), are any two trajectories of the system corresponding to the initial conditions x₀ = 0.

We then have the following lemma [236].

Lemma B.0.2 Consider the nonlinear system (B.1) together with the integral functionals (B.2) satisfying the Assumptions (a1)-(a3). Then J₀(x(.), w(.)) ≥ 0 ∀{x(.), w(.)} ∈ X × W subject to J_i(x(.), w(.)) ≥ 0, i = 1,…, l if, and only if, there exist a set of numbers τ_i ≥ 0, i = 0,…, l, ${\mathrm{\Sigma }}_{i=0}^l{\tau }_i>0$ such that

${\tau }_0{J}_0\left(x(.),w(.)\right)\ge {\displaystyle \sum _{i=1}^i{\tau }_i{J}_i\left(x(.),w(.)\right)}$

for all {x(.), w(.)} ∈ X × W, x(.) a trajectory of (B.1).

We now present the proof of the Theorem.

Combine the system (8.25) and filter (8.26) dynamics into the augmented system:

where v ∈ W is an auxiliary disturbance signal that excites the uncertainties. The objective is to render the following functional:

Let J₁(x(.), ξ(.), v(.)) represent also the integral quadratic constraint:

$J_1\left(x(.),\xi (.),\upsilon (.)\right)={\displaystyle {\int }_0^{\infty }\left({\Vert E\left(x\right)\Vert }^2-{\Vert \upsilon \left(t\right)\Vert }^2\right)dt\ge 0.}$

Then, by Lemma B.0.2, the Assumption (a1) holds for all x(.), w(.) and v(.) satisfying (a2) if, and only if, there exists numbers τ₀, τ₁, τ₀ + τ₁ > 0, such that

${\tau }_0{J}_0(.)-{\tau }_1{J}_1(.)\ge 0\forall w(.),v(.)\in W$

for all x(.), ξ(.) satisfying (a1). It can be shown that both τ₀ and τ₁ must be positive by considering the following two cases:

1.  τ₁ = 0. Then τ₀ > 0 and J₀ ≥ 0. Now set w = 0 and by (a3) we have that $\tilde{z}=0$ ∀t ≥ 0 and v(.). However this contradicts Assumption (a2), and therefore τ₁ > 0.

2.  τ₀ = 0. Then J₁ < 0. But this immediately violates the assertion of the lemma. Hence, τ₀ = 0.

Therefore, there exists τ > 0 such that

We now prove the necessity of the Theorem.

(Necessity:) Suppose (8.27) holds for the augmented system (B.3), i.e.,

Then we need to show that

for the scaled system. Accordingly, let

$w_s=0\forall t>T$

without any loss of generality. Then choosing

$\left[\begin{array}{l}w(.)\\ \tau \upsilon (.)\end{array}\right]=w_s$

we seek to make the trajectory of (B.3) identical to that of (8.29) with the filter (8.30). The result now follows from (B.7).

(Sufficiency:) Conversely, suppose (B.9) holds for the augmented system (B.3). Then we need to show that (B.8) holds for the system (8.25). Indeed, for any w, v ∈ W, we can assume w(t) = 0 ∀t > T without any loss of generality. Choosing now

$w_s=\left[\begin{array}{l}w(.)\\ \tau \upsilon (.)\end{array}\right]\forall t\in \left[0,T\right]$

and w_s(t) = 0 ∀t > T. Then, (B.9) implies (B.7) and hence (B.4). Since w(.) is truncated, we get (B.8). □