We begin with a preliminary lemma.
Consider the nonlinear time-varying system:
(B.1) |
where x(t) ∈ℜn is the state of the system and w(t) ∈ℜr is an input, together with the cost functional:
(B.2) |
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:
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 Ji(x(.), w(.)) ≥ 0, i = 1,…, l if, and only if, there exist a set of numbers τi ≥ 0, i = 0,…, l, such that
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:
(B.3) |
where v ∈ W is an auxiliary disturbance signal that excites the uncertainties. The objective is to render the following functional:
(B.4) |
Let J1(x(.), ξ(.), v(.)) represent also the integral quadratic constraint:
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
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 ∀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
(B.5) |
(B.6) |
(B.7) |
We now prove the necessity of the Theorem.
(Necessity:) Suppose (8.27) holds for the augmented system (B.3), i.e.,
(B.8) |
Then we need to show that
(B.9) |
for the scaled system. Accordingly, let
without any loss of generality. Then choosing
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
and ws(t) = 0 ∀t > T. Then, (B.9) implies (B.7) and hence (B.4). Since w(.) is truncated, we get (B.8). □