Metodolog´ıa y configuraci´ on en la recolecci´ on de datos

The main result of this section is given by Theorem 4.33, which is also presented in two different frameworks: one requires the well-posedness of the perturbed closed-loop system; while the other one requires only the uniqueness property of the perturbed closed-loop system.

The following assumptions on the normed vector space _𝒲+ are only required in the proof of Theorem4.33 with condition II:

Assumption 4.32. (1) For any 𝑥∈ 𝒲+

𝑒 , if ∥𝑥∥ < ∞, then 𝑥 ∈ 𝒲+; (2) The normed

vector space _𝒲+ _{(not necessarily complete) is truncation complete, i.e., 𝒲[0, 𝜏) is com-}

plete for any 0 < 𝜏 < _{∞; (3) For any time interval 𝐽 ≜ [0, 𝜏) with 0 < 𝜏 < ∞,} there exists a continuous map 𝐸𝐽 : 𝒲(𝐽) → 𝒲+ such that 𝑅𝐽𝑥 = 𝑅𝐽(𝐸𝐽𝑥) for any

𝑥_{∈ 𝒲(𝐽).}

Theorem 4.33. Assume that 𝑃 , ˜𝑃 , and 𝐶 are well-posed and causal systems, and that [𝑃, 𝐶] is time-invariant, well-posed and causal, and that [ ˜𝑃 , 𝐶] is causal. Let [𝑃, 𝐶] be locally input to output stable, i.e., there exist 𝑑 > 0 and functions 𝛽∈ 𝒦ℒ and 𝛾 ∈ 𝒦∞

such that, _∀𝑥0 = (𝑥10, 𝑥20)∈ 𝔖𝑃 × 𝔖𝐶, ∀𝑤0+∈ 𝒲+, ∀𝑡 ≥ 0,

max_{𝜒(𝑥0),∥𝑤0+∥} ≤ 𝑑 ⇒ ∣(Π𝑥_𝑃//𝐶0 𝑤0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑥0), 𝑡) + 𝛾(∥𝑤0+∥[0,𝑡)), (4.77)

If there exist functions 𝜎0, 𝜎 ∈ 𝒦∞ and 𝛽0 ∈ 𝒦ℒ such that for any ˜𝑤1− ∈ 𝒲−∩ 𝔅−_𝑃˜

there exists a 𝑤_{1− ∈ 𝒲}−_{∩ 𝔅}− 𝑃 with

∥𝑤1−∥ ≤ 𝜎0(∥ ˜𝑤1−∥), (4.78)

and a causal surjective operator Φ : dom(Φ) _{⊆ 𝒢}𝑤1−

𝑃 → 𝒢 ˜ 𝑤1− ˜ 𝑃 satisfying, ∀𝑡 > ℎ ≥ 0, _∀𝑤1+ ∈ dom(Φ) with ∥𝑤1+∥ ≤ 𝛽(𝑑, 0) + 𝛾(𝑑), ∣((Φ − 𝐼)𝑤1+)(𝑡)∣ ≤ 𝛽0(∥𝑤1−∧𝑤1+∥_(−∞,ℎ], 𝑡− ℎ) + 𝜎(∥𝑤1+∥_[ℎ,𝑡)). (4.79)

In addition, if there exist two functions 𝜌, 𝜀 of class _𝒦∞ such that, _{∀𝑠 ≥ 0,}

𝜎∘ (𝐼 + 𝜌) ∘ 𝛾(𝑠) ≤ (𝐼 + 𝜀)−1(𝑠). (4.80) And either of the following conditions is satisfied:

I. [ ˜𝑃 , 𝐶] is well-posed and Π𝑥˜0

˜ 𝑃 //𝐶(𝒲

+_{)⊆ 𝒲}+ _{for any ˜𝑥}

0 ∈ 𝔖_𝑃˜× 𝔖𝐶;

II. Assumption4.32holds for_𝒲+, and [ ˜𝑃 , 𝐶] has the uniqueness property, and Π𝑥0

𝑃//𝐶

is relatively continuous for any 𝑥0 ∈ 𝔖𝑃 × 𝔖𝐶, and 𝑅[0,𝜏 )(Φ− 𝐼) is compact for

any 0 < 𝜏 <_∞.

Then the closed-loop system [ ˜𝑃 , 𝐶] is also locally input to output stable. More specifically, there exist ˜𝑑 > 0, for any function 𝛼 of class _𝒦∞, there exists a function ˜𝛽 _{∈ 𝒦ℒ such} that, ∀˜𝑥0 ∈ 𝔖_𝑃˜ × 𝔖𝐶, ∀ ˜𝑤0+ ∈ 𝒲+, ∀𝑡 > 0,

max_{𝜒(˜𝑥0),∥ ˜𝑤0+∥} ≤ ˜𝑑⇒ ∣(Π𝑥_{𝑃 //𝐶}˜_˜0 𝑤˜0+)(𝑡)∣ ≤ ˜𝛽 (𝜒(˜𝑥0), 𝑡) + (𝛼 + ˜𝛾)(∥ ˜𝑤0+∥_[0,𝑡)) (4.81)

where ˜𝑑 = min_{{(𝐼 + Δ)}−1_{∘ (𝐼 + 𝜀}−1₎−1_{(𝑑), (𝜎0+ 𝐼)}−1_{(𝑑)} with functions Δ ∈ 𝒦 and}

˜

𝛾 ∈ 𝒦∞ defined by

Δ(𝑟) ≜ 𝛽0((𝜎0+ 𝐼)(𝑟), 0) + 𝜎∘ (𝐼 + 𝜌−1)∘ 𝛽((𝜎0+ 𝐼)(𝑟), 0), ∀𝑟 ≥ 0, (4.82a)

˜

𝛾(𝑟) ≜ (𝜎 + 𝐼)_{∘ (𝐼 + 𝜌) ∘ 𝛾 ∘ (𝐼 + 𝜀}−1)3(𝑟), _{∀𝑟 ≥ 0.} (4.82b) Proof. To prove above theorem we need to change slightly the proof of Theorem4.8in Chapter 4. Choose ˜𝑑 = min_{{(𝐼 + Δ)}−1_{∘ (𝐼 + 𝜀}−1₎−1_{(𝑑), (𝜎0+ 𝐼)}−1_{(𝑑)}. Note that the}

function Δ defined in (4.82) is the same as (4.14) (or (4.27)) in the proof of Theorem

4.8. For any max{𝜒(˜𝑥0),∥ ˜𝑤0+∥} ≤ ˜𝑑, from (4.8) (or (4.24)) and ˜𝑑≤ (𝜎0+ 𝐼)−1(𝑑) we

have 𝜒(𝑥0)≤ 𝑑; and from ˜𝑑≤ (𝐼 + Δ)−1∘ (𝐼 + 𝜀−1)−1(𝑑) and (4.13) we have∥𝑤0+∥ ≤ 𝑑.

The rest of proof follows from the proof of Theorem4.8on page 81.

4.8

Summary

In Chapter 3 we have developed a unified construction of an underlying abstract state space applicable to input-output systems defined over a doubly infinite time axis. The current chapter is the main part of this thesis, which provides an input-output theory with an integrated treatment of initial conditions, culminating in a statement and proof of a robust stability result. The resulting gap distances take into account both the effect of the perturbation on the state space structure (and hence the initial condition) as well as the input-output response. This complements the robust stability theory of Georgiou and Smith [Georgiou and Smith,1997b] by introducing initial conditions and applies the

ideas of the ISS framework in a situation whereby the conventional state-space formalism of ISS is not directly applicable due to variation in the structure of the state space between the nominal and perturbed systems which arise naturally in a robust stability setting. Two different versions of the main results are presented. One requires the well- posedness, while the other one requires only the uniqueness property of the perturbed closed-loop system. In real-world applications, both well-posedness (i.e., existence and uniqueness) and stability are required to be verified for a feedback system. In general uniqueness conclusions are more easily obtained than existence conclusions. Establishing existence and stability simultaneously by only using uniqueness greatly eases the real- time application of the robust stability result (see also the discussions given in [French and Bian,2012]). Generalisation of this robust stability result to systems with potential for finite escape times is discussed at the end of this chapter.

can lead us further and not accumulation of facts.

Albert Einstein (1879-1955)

Chapter 5

Generalised Small-Gain Theorem

for Systems with Initial

Conditions

5.1

Introduction

The use of the small-gain theorem in control theory dates back to the 1960’s by [Zames,

1966b,c] and [Sandberg,1964]. The original version of the small-gain theorem involves systems with finite linear gains from input to output with or without a bias term (see e.g., [Desoer and Vidyasagar,1975]). Extensions of the small-gain theorem to nonlinear gains have been studied by many researchers. The work on the small-gain theory involving nonlinear gain began with [Hill, 1991, Mareels and Hill, 1992], where the monotone gain was proposed for a nonlinear generalisation of the classical small-gain theorem. In [Jiang et al., 1994], the authors developed a nonlinear ISS-type small-gain theorem in the sense of [Sontag, 1989] for interconnection of nonlinear systems in state space representations, which led an extensive follow-up literature (e.g., [Chen and Huang,

2005, Jiang and Marcels, 1997, Jiang et al., 1996]). Several interesting extensions of the small-gain theorem were also obtained for systems with special structures such as Volterra systems [Zheng and Zafiriou,1999], general networks [Dashkovskiy et al.,2007], large-scale complex systems [Jiang and Wang,2008], stochastic systems [Lu and Skelton,

2002], hybrid systems [Liberzon and Neˇsi´c, 2006, Neˇsi´c and Teel, 2008], etc. In the present chapter, we present a nonlinear small-gain theorem on input to output stability for nonlinear feedback systems from an input-output point of view.

Note that the classical small-gain theorem obtained in the input-output framework has the benefit that the stability property is completely disconnected from the existence, uniqueness property, etc.; see e.g., [Desoer and Vidyasagar, 1975]. Most of the results of the ISS-type nonlinear small-gain theorem were obtained for nonlinear state space

𝐺 𝐻 𝑢0 𝑢1 𝑢2 𝑦1 𝑦2 ? 𝑦0 - 6 - 

Figure 5.1: Nonlinear feedback configuration [𝐺, 𝐻]

models, and a priori requirements of existence and uniqueness property of systems are imposed (e.g., requiring smoothness or Lipschitz continuity of dynamical functions), and extra “observability” conditions are imposed to guarantee that the state trajectories are bounded when the input and output are bounded. In [Ingalls et al.,1999, Sontag and Ingalls,2002], the authors presented an abstract ISS-type small-gain theorem including applications to purely input/output systems represented by i/o operators defined on spaces of signals beginning at some finite time in the past. The special representation of systems allows the authors to identify the ‘state’ only with the past input without using the past output; but it precludes for example the uncontrollable stable linear case (see also the discussion related to Theorem4.8 on page81).