About the Stabilization of a Nonlinear Perturbed Difference Equation

Abstract

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order than that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e., perturbation-free) difference equation plus an incremental controller which completes the stabilization in the presence of perturbed or unmodeled dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize both the nominal difference equation and also the perturbed one under a small-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle, and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.

1. Introduction

In this paper, the following nonautonomous difference equation is investigated:𝑥_𝑛=ℎ_𝑛𝑥_𝑛−1,…,𝑥_𝑛−𝑚=𝑓_𝑛𝑥_𝑛−1,…,𝑥_𝑛−𝑚0+𝑓^∼_𝑛𝑥_𝑛−1,…,𝑥_𝑛−^∼_𝑚+𝑔_𝑛𝑥_𝑛−1,…,𝑥_{𝑛−𝑚𝑔}+𝑔^∼_𝑛𝑥_𝑛−1,…,𝑥_𝑛−^∼_𝑚𝑔=𝑥⁰_𝑛+𝑥^∼_𝑛+𝑥^𝑐_𝑛+𝑥^∼𝑐_𝑛;𝑛∈𝐍, (1.1) of order 𝑚∶=max(𝑚₀,𝑚^∼,𝑚_𝑔,𝑚^∼_𝑔)≥1 and initial conditions 𝑥_1−𝑚,…,𝑥₀, where the four terms of the second identity are pair-wise identical in the same order as written, in which ℎ_𝑛∶𝐷⊂𝐑^𝑚→𝐑, 𝑓_𝑛∶𝐷_𝑓⊂𝐑^𝑚0→𝐑, 𝑓^∼_𝑛∶𝐷^∼_𝑓⊂𝐑^∼𝑚→𝐑, 𝑔_𝑛∶𝐷_𝑔⊂𝐑^𝑚𝑔→𝐑, and 𝑔^∼_𝑛∶𝐷^∼_𝑔⊂𝐑^∼𝑚𝑔→𝐑; 𝐷 is a nonempty subset of the union of the sets 𝐷_𝑓, 𝐷^∼_𝑓, 𝐷_𝑔, and 𝐷^∼_𝑔. The four pair-wise identical terms of the last identity have the following interpretations:(i)𝑥⁰_𝑛 is the nominal value of the uncontrolled nominal solution 𝑥_𝑛 at the 𝑛th sample in the absence of perturbations and controls;(ii)𝑥^∼_𝑛 is the perturbed uncontrolled solution which can be generated for perturbed parameterizations (then 𝑚^∼≤𝑚₀) and possibly contributed by unmodeled dynamics (then 𝑚^∼>𝑚₀);(iii)𝑥^c_𝑛 is the correction by some nominal feedback controller of the uncontrolled nominal solution which can be potentially used to stabilize it or to improve it in some practical suitable sense provided it is already stable;(iv)𝑥^∼𝑐_𝑛 is the correction by adding some incremental feedback controller of the perturbed nominally controlled solution;(v)𝐍 is the set of natural numbers, and 𝐍₀=𝐍∪{0} is that of nonnegative integers;(vi)𝑆⁰ and cl𝑆 denote, respectively, the interior and closure of the set 𝑆.

The stability and instability properties of nonlinear difference equations have been investigated in a set of papers. See, for instance, [1–13] and references therein. There is a wide set of problems where stability of discrete systems involving either the discretization of time-continuous systems or being essentially digital by nature are of interest and involving very often the presence of nonlinearities. In those problems stability is commonly a required property to be well posed. Among such problems, we can mention (a) those related to signal processing, (b) models involving neural networks, (c) adaptive control to deal with not perfectly known systems under combined estimation and control, (d) problems related to modelling dynamic systems describing biological, medical, or ecological systems, and (e) those related to descriptions to rational difference equations. See, for instance, [14–20] and references therein. Note that the nominal uncontrolled particular case of (1.1) is given by the constraint 𝑓^∼_𝑛+𝑔_𝑛+𝑔^∼_𝑛≡0 for all 𝑛∈𝐍. The main objective of this paper is the study of the stability and instability of equilibrium points of a very general nonlinear autonomous difference equations which include additive perturbations. There is no essential formal distinction through the paper between parametrical perturbations, or structured and unstructured unmodeled dynamics except that each of them has its own description part in the relevant formulas [21–23]. Note that parametrical perturbations do not modify the order of the nominal equation, while unmodeled dynamics increases such an order. There are abundant examples in nature where unmodeled dynamics is inherently present as, for instance, the problem of the antimissile/missile targeting process under fast attack/defence manoeuvres, which generates high-frequency signals, or the parasite capacitors between an electronic amplifier and ground for high-frequency exciting signals. The perturbation-free difference equation will be referred to as the nominal uncontrolled one, while the perturbed difference equation will be referred to as the uncontrolled perturbed difference equation. Two classes of feedback controllers are also proposed to stabilize the uncontrolled autonomous difference equation. The first class consists of two additive dynamics, namely, the nominal control for stabilization of the uncontrolled nominal equation plus an incremental controller for stabilization of the unmodeled dynamics. The second class consists of a single controller which stabilizes the whole uncontrolled dynamics for a certain tolerance to presence of perturbation dynamics of sufficiently small size characterized in terms of sufficiently small norm. The perturbed uncontrolled difference equation and the controlled difference equation can potentially possess distinct equilibrium points than the uncontrolled nominal difference equation. The formalism can also be applied to the study of feedback stabilization of unstable oscillations.

2. Vectorization Preliminaries and Linearization-Based Relations between Equilibrium Points and Limit Oscillatory Solutions

Problems of major interest concerning (1.1) are (a) the characterization of a controller which stabilizes, at least locally around an equilibrium point, an unstable nominal difference equation and (b) the stabilization of either a particular or a class of perturbed uncontrolled equations under a combined nominal plus incremental controller. It has to be pointed out that any equilibrium point of the uncontrolled equations can be reallocated under a control action. In other words, the local stabilization via feedback control of an unstable equilibrium point of the uncontrolled equations may lead in parallel to a reallocation of such an equilibrium point. An associate vector function to (1.1) of dimension 𝑚 is𝑉_ℎ𝑢₁,…,𝑢_𝑚=ℎ𝑢₁,…,𝑢_𝑚,𝑢₁,…,𝑢_𝑚−1=𝑉_𝑓𝑢₁,…,𝑢_𝑚+𝑉^∼_𝑓𝑢₁,…,𝑢_𝑚+𝑉_𝑔𝑢₁,…,𝑢_𝑚+𝑉^∼_𝑔𝑢₁,…,𝑢_𝑚=𝑓𝑢₁,…,𝑢_𝑚0+𝑓^∼𝑢₁,…,𝑢^∼_𝑚+𝑔𝑢₁,…,𝑢_𝑚𝑔+𝑔^∼𝑢₁,…,𝑢^∼_𝑚𝑔,𝑢₁,…,𝑢_𝑚−1, (2.1) where𝑉_𝑓𝑢₁,…,𝑢_𝑚∶=𝑓𝑢₁,…,𝑢_𝑚0,𝑢₁,…,𝑢_𝑚−1,𝑉^∼_𝑓𝑢₁,…,𝑢_𝑚∶=𝑓^∼𝑢₁,…,𝑢^∼_𝑚,𝑢₁,…,𝑢_𝑚−1,𝑉_𝑔𝑢₁,…,𝑢_𝑚∶=𝑔𝑢₁,…,𝑢_𝑚𝑔,𝑢₁,…,𝑢_𝑚−1,𝑉^∼_𝑔𝑢₁,…,𝑢_𝑚=𝑔^∼𝑢₁,…,𝑢_𝑚∼𝑔,𝑢₁,…,𝑢_𝑚−1. (2.2) In particular, 𝑉_ℎ(𝑢₁)=(ℎ(𝑢₁),𝑢₁) if 𝑚=1, one has the following particular case of  (2.2)𝑉_𝑓𝑢₁∶=𝑓𝑢₁;𝑉^∼_𝑓𝑢₁∶=𝑓^∼𝑢₁;𝑉_𝑔𝑢₁∶=𝑔𝑢₁;𝑉^∼_𝑔𝑢₁∶=𝑔^∼𝑢₁. (2.3) The following result follows by simple inspection of (2.1) since 𝑚≥𝑚₀, that is, the dimension of the current difference equation is not less than that of its nominal version. Note that, if 𝑚=𝑚₀, then the current difference equation has the same dimensionality as that of its nominal counterpart as discussed for such a case in the formalism proposed and developed in [12].

Lemma 2.1. The vector function (2.1) can be expressed equivalently as 𝑉_ℎ𝑢₁,…,𝑢_𝑚=𝑓𝑢₁,…,𝑢_𝑚0,𝑚−𝑚₀0,…,0,𝑢₁,…,𝑢_𝑚−1+𝑓^∼𝑢₁,…,𝑢^∼_𝑚+𝑔𝑢₁,…,𝑢_𝑚𝑔+𝑔^∼𝑢₁,…,𝑢^∼_𝑚𝑔,0,𝑚−1…,0, (2.4) where 𝑓∶(𝐷_𝑓∪𝐷^∼_𝑓)⊂𝐑^𝑚→𝐑 is defined from 𝑓∶𝐷_𝑓⊂𝐑^𝑚0→𝐑 by adding (𝑚−𝑚₀) identically zero arguments. The set 𝐷 can be identical (although it is non-necessarily identical) to 𝐷_𝑓 only if 𝑚=𝑚₀, and then there is a unique such a mapping which is the identity self-mapping.

The nominal and perturbed uncontrolled difference equations as well as the nominal controlled and perturbed controlled ones can have potentially distinct equilibrium points as follows. A generic “ad hoc” description is also useful to describe some limit oscillatory solutions:(1)𝑥⁰ is an equilibrium point of the uncontrolled nominal difference equation 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0) if and only if 𝑥⁰=𝑓_𝑛(𝑥⁰,…,𝑥⁰) for all 𝑛∈𝐍. Then, 𝑋⁰=(𝑥⁰,…,𝑥⁰) is the associate equilibrium point of the first-order autonomous 𝑚₀-order vector equation𝑋_𝑛=𝑉_𝑓𝑛(𝑋_𝑛−1); for all 𝑛∈𝐍 obtained from the particular difference equation (1.1) 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0) for all 𝑛∈𝐍 via the nominal vector equation 𝑉_𝑓𝑛(𝑢₁,…,𝑢_𝑚)=(𝑓(𝑢₁,…,𝑢_𝑚0),𝑢₁,…,𝑢_𝑚−1) provided that 𝑉_𝑓𝑛(𝐷_𝑓)⊆𝐷_𝑓. A sequence solution (𝑥⁰₁,…,𝑥⁰_𝑚0) of 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0) is a limit oscillatory solution of order at most 𝑚₀ if and only if 𝑥⁰_{𝑘𝑚0+𝑖}=𝑓_𝑛(𝑥⁰₁,…,𝑥⁰_𝑚0) for all 𝑘,𝑛∈𝐍 for all 𝑖∈𝑚₀∶={1,2,…,𝑚₀}. Such a solution is trivially an equilibrium point if 𝑥⁰_𝑖=𝑥⁰ for all 𝑖∈𝑚₀. The 𝑚₀-real vector 𝑋⁰=(𝑥⁰₁,…,𝑥⁰_𝑚0) is the associate nominal limit oscillatory solution of order at most 𝑚₀ of the first-order autonomous 𝑚₀-order vector equation 𝑋_𝑛=𝑉_𝑓𝑛(𝑋_𝑛−1) for all 𝑛∈𝐍 obtained from the particular difference equation (1.1) 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0) for all 𝑛∈𝐍 via the nominal vector equation 𝑉_𝑓𝑛(𝑢₁,…,𝑢_𝑚)=(𝑓(𝑢₁,…,𝑢_𝑚0),𝑢₁,…,𝑢_𝑚−1);(2)𝑥^0𝑝 is an equilibrium point of the uncontrolled perturbed difference equation 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0)+𝑓^∼_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−^∼_𝑚) if and only if 𝑥^0𝑝=𝑓_𝑛(𝑥^0𝑝,…,𝑥^0𝑝)+𝑓^∼_𝑛(𝑥^0𝑝,…,𝑥^0𝑝) for all 𝑛∈𝐍. Then, 𝑋^0𝑝=(𝑥^0𝑝,…,𝑥^0𝑝) is the associate equilibrium point of the first-order autonomous 𝑚_0𝑝∶=max(𝑚₀,𝑚^∼)-order vector equation 𝑋_𝑛=𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑋_𝑛−1) for all 𝑛∈𝐍 according to: 𝑉_𝑓+^∼_𝑓𝑢₁,…,𝑢_𝑚0𝑝=𝑓₀𝑢₁,…,𝑢_𝑚0𝑝,𝑚_0𝑝−𝑚₀0,…,0,𝑢₁,…,𝑢_{𝑚0𝑝−1}+𝑓^∼₀𝑢₁,…,𝑢^∼_𝑚,𝑚_0𝑝−𝑚^∼,𝑚0,…,0^0𝑝−10,…,0, (2.5) provided that 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝐷_𝑓∪𝐷^∼_𝑓)⊆𝐷_𝑓∪𝐷^∼_𝑓  provided the set union is nonempty, where 𝑓₀,𝑓^∼∶(𝐷_𝑓∪𝐷^∼_𝑓)⊂𝐑^𝑚0𝑝→𝐑 in (2.5), provided that 𝐷_𝑓∪𝐷^∼_𝑓  is non-empty, take into account that the uncontrolled and nominal perturbed difference equations have potentially distinct orders and are built from𝑓∶𝐷_𝑓⊂𝐑^𝑚0→𝐑, 𝑓^∼∶𝐷^∼_𝑓⊂𝐑^∼𝑚→𝐑 as in the parallel construction of Lemma 2.1, (2.4). We can describe limit oscillatory solutions of the uncontrolled perturbed difference equation of order at most 𝑚_0𝑝, or equivalently those of its associate vector function, by a sequence solution: 𝑥^0𝑝_{𝑘𝑚0𝑝+𝑖}=𝑓_𝑛𝑥₁^0𝑝,…,𝑥_𝑚^0𝑝_0𝑝+𝑓^∼_𝑛𝑥₁^0𝑝,…,𝑥_𝑚^0𝑝_0𝑝;∀𝑘,𝑛∈𝐍,∀𝑖∈𝑚_0𝑝; (2.6)(3)𝑥^𝑐 is an equilibrium point of the controlled nominal difference equation 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚0)+𝑔_𝑛(𝑥_𝑛−1,…,𝑥_{𝑛−𝑚𝑔}) if and only if 𝑥^𝑐=𝑓_𝑛(𝑥^𝑐,…,𝑥^𝑐)+𝑔_𝑛(𝑥^𝑐,…,𝑥^𝑐) for all 𝑛∈𝐍. Then, 𝑋^𝑐=(𝑥^𝑐,…,𝑥^𝑐) is the associate equilibrium point of the first-order autonomous 𝑚_𝑐∶=max(𝑚₀,𝑚_𝑔)-order vector equation 𝑋_𝑛=𝑉_{𝑓𝑛+𝑔𝑛}(𝑋_𝑛−1) for all 𝑛∈𝐍, provided that 𝑉_𝑓+𝑔(𝐷_𝑓∪𝐷_𝑔)⊆𝐷_𝑓∪𝐷_𝑔, provided that such a union is non-empty. The equivalent first-order vector equations are defined via the associate vector function 𝑉_𝑓+𝑔(𝐷_𝑓∪𝐷_𝑔) through ad hoc functions 𝑓_𝑐∶𝐷→𝐷_𝑓 and 𝑔_𝑐∶𝐷→𝐷_𝑔 built according to the corresponding associate vector equation defined in a similar way to (2.4) and (2.5). This is directly extended to limit oscillatory solutions of order at most 𝑚_𝑐 of the controlled nominal difference equation, which can be equivalently expressed in vector form, in the same way as above;(4)𝑥^𝑐𝑝 is an equilibrium point of the controlled perturbed difference equation (1.1) if and only if𝑥^𝑐𝑝=ℎ_𝑛𝑥^𝑐𝑝,…,𝑥^𝑐𝑝=𝑓_𝑛𝑥^𝑐𝑝,…,𝑥^𝑐𝑝+𝑓^∼_𝑛𝑥^𝑐𝑝,…,𝑥^𝑐𝑝+𝑔_𝑛𝑥^𝑐𝑝,…,𝑥^𝑐𝑝+𝑔^∼_𝑛𝑥^𝑐𝑝,…,𝑥^𝑐𝑝;∀𝑛∈𝐍. (2.7) Then, 𝑋^𝑐𝑝=(𝑥^𝑐𝑝,…,𝑥^𝑐𝑝) is the associate equilibrium point of the first-order autonomous 𝑚-order vector equation 𝑋_𝑛=𝑉_ℎ𝑛(𝑋_𝑛−1) for all 𝑛∈𝐍 defined via (2.4) provided that 𝑉_ℎ𝑛(𝐷)⊆𝐷. A sequence solution (𝑥₁^𝑐𝑝,…,𝑥_𝑚^𝑐𝑝) of 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚) is a limit oscillatory solution of order at most 𝑚 if and only if 𝑥^𝑐𝑝_{𝑘𝑚+𝑖}=𝑓_𝑛(𝑥₁^𝑐𝑝,…,𝑥_𝑚^𝑐𝑝) for all𝑘,𝑛∈𝐍 for all 𝑖∈𝑚. Such a solution is trivially an equilibrium point if 𝑥_𝑖^𝑐𝑝=𝑥^𝑐𝑝 for all 𝑖∈𝑚. The 𝑚-real vector 𝑋^𝑐𝑝=(𝑥₁^𝑐𝑝,…,𝑥_𝑚^𝑐𝑝) is the associate nominal limit oscillatory solution of order at most 𝑚 of the first-order autonomous 𝑚-order vector equation 𝑋_𝑛=𝑉_𝑓𝑛(𝑋_𝑛−1) for all 𝑛∈𝐍 obtained from the particular difference equation (1.1) 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚) for all 𝑛∈𝐍 via the nominal vector equation 𝑉_𝑓𝑛(𝑢₁,…,𝑢_𝑚)=(𝑓(𝑢₁,…,𝑢_𝑚),𝑢₁,…,𝑢_𝑚−1).

Remark 2.2. Note that the above description allows the characterization of equilibrium points as particular cases of limit oscillatory solutions. Note also that limiting oscillatory solutions can exceed the order of the difference equations if such a solution has a repeated pattern of more elements than the order of the difference equations. Details are omitted since the analysis method is close to the above one in both scalar and equivalent vector forms. Limiting oscillatory solutions are relevant in some applications, in particular, in the fields of communications, design of electronic oscillators, and so forth.

Remark 2.3. The difference equation 𝑥_𝑛=𝑓_𝑛(𝑥_𝑛−1,…,𝑥_𝑛−𝑚) for all 𝑛∈𝐍 has been pointed to be equivalent to its associate vector equation 𝑋_𝑛=𝑉_𝑓𝑛(𝑋_𝑛−1) for all 𝑛∈𝐍. Then, the nominal uncontrolled difference equation admits the representation 𝑥_𝑛=𝑓_𝑛(𝑋_𝑛−1)=𝑓_𝑛(𝑉_𝑓𝑛−1(𝑋_𝑛−2)) for all 𝑛∈𝐍. Proceeding recursively: 𝑥_𝑛=𝑓_𝑛𝑋_𝑛−1=𝑓_𝑛𝑉_𝑓𝑛−1𝑋_𝑛−2=𝑓_𝑛𝐺^𝑓_𝑛−1𝑋₀;∀𝑛∈𝐍. (2.8) By defining 𝐺^𝑓_𝑛∶=𝑉_𝑓𝑛o𝑉_𝑓𝑛−1o⋅⋅⋅oV_𝑓1 for all 𝑛∈𝐍₀  with 𝐺^𝑓₀  being identity. Close composed mappings to describe the various uncontrolled and controlled (nominal or) versions of (1.1) are 𝐺^𝑓+∼𝑓₀=id,𝐺^𝑓+∼𝑓_𝑛∶=𝑉_𝑓𝑛+^∼_𝑓𝑛o𝑉_{𝑓𝑛−1+}^∼_𝑓𝑛−1o⋅⋅⋅oV_𝑓1+^∼_𝑓1𝐺;∀𝑛∈𝐍,_{𝑓+∼𝑓0+𝑔}=id,𝐺^𝑓+∼𝑓_𝑛^+𝑔∶=𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}o𝑉_{𝑓𝑛−1+}^∼_{𝑓𝑛−1+𝑔𝑛−1}o⋅⋅⋅oV_𝑓1+^∼_𝑓1+𝑔1𝐺;∀𝑛∈𝐍,_ℎ0=id,𝐺^ℎ_𝑛∶=𝑉_ℎ𝑛o𝑉_ℎ𝑛−1o⋅⋅⋅oV_ℎ1;∀𝑛∈𝐍. (2.9)

The following result is direct by inspection of (1.1).

Proposition 2.4. The nominal and perturbed uncontrolled associate vector functions may have a common equilibrium point or a common limiting oscillation of order at most m only if 𝑚=𝑚₀=max(𝑚₀,𝑚^∼). The nominal and perturbed uncontrolled associate vector functions as well as the controlled and perturbed controlled ones may have a common equilibrium point only if, in addition, 𝑚₀=max𝑚₀,𝑚_𝑔,𝑚^∼_𝑔=max𝑚₀,𝑚^∼_𝑔. (2.10)

Proof. If the conditions fail and the vector functions referred to have some common equilibrium point, this one, should have different dimension depending on the equation, which is a contradiction. The proof is also valid “mutatis-mutandis” for limiting oscillation of orders at most 𝑚.

It is now discussed the presence of limit oscillations of the uncontrolled perturbed difference equations in a neighbourhood centred about a nominal limit oscillation. A similar analysis is also useful for closeness of the limiting oscillatory solutions to that of a given difference equation of any of the three remaining difference ones under investigation.

Theorem 2.5. Assume the following:(1)𝑚_𝑝∶=max(𝑚₀,𝑚^∼)=𝑚₀, 𝑉_𝑓𝑛(𝐷_𝑓)⊆𝐷_𝑓≠∅ and 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝐷_𝑓∪𝐷^∼_𝑓)⊆𝐷_𝑓∪𝐷^∼_𝑓≠∅ for all 𝑛∈𝐍, where 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑢₁,…,𝑢_𝑚𝑝) is defined in (2.5) for all 𝑛∈𝐍;(2)𝜕𝑓_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, 𝜕𝑓^∼_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰ exist within a neighborhood of 𝑋⁰, which is a limit oscillatory solution of the vector uncontrolled nominal equation of order at most 𝑚_𝑝=𝑚₀ (including potentially nominal equilibrium points as particular cases); (3)the inverse 𝑚₀-matrix (𝐼_𝑚𝑝−𝑀⁰(𝑋⁰))⁻¹ exists, where 𝑀⁰(𝑋⁰)=𝜕𝑉_𝑓(𝑋)/𝜕𝑋^𝑇|_𝑋⁰+𝜕𝑉^∼_𝑓(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, and 𝐼_𝑚0 is the 𝑚₀-identity matrix of 𝐑^𝑚0. Then, 𝑋^0𝑝=𝑋⁰+(𝐼_𝑚−𝑀⁰(𝑋⁰))⁻¹𝑉^∼_𝑓(𝑋⁰) is a linear estimate of limit oscillatory solutions of order at most 𝑚₀ (including, as particular cases, potential equilibrium points) of the vector perturbed difference equation. The estimate is closed to the true values of 𝑋^0𝑝 as ‖𝑉^∼_𝑓(𝑋⁰)‖ is sufficiently small.
If rank(𝐼_𝑚𝑝−𝑀⁰(𝑋⁰),𝑉^∼_𝑓(𝑋⁰))=rank(𝐼_𝑚𝑝−𝑀⁰(𝑋⁰))<𝑚_𝑝, then there are infinitely many first-order estimates 𝑋^0𝑝 of limiting oscillatory solutions of the vector uncontrolled nominal equation of order at most 𝑚₀. If 𝑚_𝑝−1≥rank(𝐼_𝑚−𝑀⁰(𝑋⁰)and𝑉^∼_𝑓(𝑋⁰))>rank(𝐼_𝑚𝑝−𝑀⁰(𝑋⁰)), then there is no such an estimate.

Proof. Note that 𝑚_𝑝=𝑚₀ implies that 𝑚^∼≤𝑚₀. Define Δ𝑋^0𝑝,0∶=𝑋^0𝑝−𝑋⁰ which is rewritten below after using a linearized perturbed difference vector equation since the perturbed equilibrium point is within a neighbourhood of the nominal one: Δ𝑋^0𝑝,0_𝑛+1=𝑀⁰_𝑛𝑋⁰Δ𝑋_𝑛^0𝑝,0+𝑉^∼_𝑓𝑛𝑋⁰+𝑜Δ𝑋_𝑛^0𝑝,0𝐼_𝑚𝑝=𝑀⁰𝑋⁰Δ𝑋_𝑛^0𝑝,0+𝑉^∼_𝑓𝑋⁰+𝑜Δ𝑋_𝑛^0𝑝,0𝐼_𝑚𝑝;∀𝑛∈𝐍, (2.11) where 𝑀⁰_𝑛𝑋⁰=𝜕𝑉_𝑓𝑛(𝑋)𝜕𝑋^𝑇_𝑋⁰+𝜕𝑉^∼_𝑓𝑛(𝑋)𝜕𝑋^𝑇_𝑋0=𝜕𝑓_𝑛(𝑋)𝜕𝑋^𝑇_𝑋⁰+𝜕𝑓^∼_𝑛(𝑋)𝜕𝑋^𝑇_𝑋0𝐼_𝑚𝑝−10=𝑀⁰𝑋⁰=𝜕𝑉_𝑓(𝑋)𝜕𝑋^𝑇_𝑋⁰+𝜕𝑉^∼_𝑓(𝑋)𝜕𝑋^𝑇_𝑋0=𝜕𝑓(𝑋)𝜕𝑋^𝑇_𝑋⁰+𝜕𝑓^∼(𝑋)𝜕𝑋^𝑇_𝑋0𝐼_𝑚−10;∀𝑛∈𝐍, (2.12) if 𝑚₀≥2, where 𝐼_𝑚0−1 is the (𝑚₀−1) identity matrix and superscript 𝑇 denotes transposition, since 𝑉_𝑓𝑛(𝑋⁰)=𝑉_𝑓(𝑋⁰), 𝑉^∼_𝑓𝑛(𝑋⁰)=𝑉^∼_𝑓(𝑋⁰), 𝑓_𝑛(𝑋⁰)=𝑓(𝑋⁰), 𝑓^∼_𝑛(𝑋⁰)=𝑓^∼(𝑋⁰) for all 𝑛∈𝐍; 𝑀⁰_𝑛𝑋⁰=𝜕𝑉_𝑓𝑛(𝑋)𝜕𝑋_𝑋⁰+𝜕𝑉^∼_𝑓𝑛(𝑋)𝜕𝑋_𝑋0=𝑀⁰𝑋⁰=𝜕𝑉_𝑓(𝑋)𝜕𝑋_𝑋⁰+𝜕𝑉^∼_𝑓(𝑋)𝜕𝑋_𝑋0;∀𝑛∈𝐍, (2.13) if 𝑚₀=1. Taking Δ𝑋^0𝑝,0_𝑛+1=Δ𝑋_𝑛^0𝑝,0=Δ𝑋^0𝑝,0 for all 𝑛∈𝐍, one gets from (2.11): Δ𝑋^0𝑝,0=𝐼_𝑚0−𝑀⁰𝑋⁰⁻¹𝑉^∼_𝑓𝑋₀+𝑜Δ𝑋^0𝑝,0𝐼_𝑚0, (2.14) provided that (𝐼_𝑚0−𝑀⁰(𝑋⁰))⁻¹ exists so that Δ𝑋^0𝑝,0=(𝐼_𝑚−𝑀⁰(𝑋⁰))⁻¹𝑉^∼_𝑓(𝑋₀) is an estimate of Δ𝑋^0𝑝,0_𝑛+1 so that if ‖𝑀⁰(𝑋⁰)‖<1 and ‖𝑉^∼_𝑓(𝑋₀)‖<(1−‖𝑀⁰(𝑋⁰)‖)𝜀 for some 𝜀∈𝐑₊, then one gets from Banach’s perturbation lemma [24]: Δ𝑋^0𝑝,0≤𝐼_𝑚−𝑀⁰𝑋⁰⁻¹𝑉^∼_𝑓𝑋₀≤𝑉^∼_𝑓𝑋⁰1−𝑀⁰𝑋⁰<𝜀. (2.15) Since for 𝜀=0, 𝑉^∼_𝑓(𝑋₀)=Δ𝑋^0𝑝,0=0 then, for a sufficiently small 𝜀^∗ such that ‖𝑉^∼_𝑓(𝑋⁰)‖/(1−‖𝑀⁰(𝑋⁰)‖)<𝜀^∗ and for any 𝜀≤𝜀^∗, 𝑜(‖Δ𝑋^0𝑝,0‖)≤𝜀/2 what occurs in particular, for 𝜀<1 if 𝑓,𝑓^∼∶𝐑^𝑚→𝐑 are furthermore analytic in an open ball of 𝐑^𝑚 centred at 𝑋₀ of radius 𝜌=3𝜀/2. The conditions for the existence of infinitely many first-order estimates of Δ𝑋^0𝑝,0 or the existence of none of them is direct from compatible and incompatible conditions for linear algebraic systems of equations according to Rouché-Froebenius theorem from linear algebra.

Note that it can occur for the nominal and perturbed uncontrolled difference equations to have common equilibrium points. On the other hand, it is possible to obtain linear similar first-order comparison results to those of Theorem 2.5 for the estimates of the equilibrium points of the corrected closed-loop system via an incremental controller related to those of the controlled system without incremental controller. An “ad hoc” result is now stated without proof which can be performed very closely to that of Theorem 2.5:

Theorem 2.6. Assume the following: (1)𝑚=max(𝑚₀,𝑚^∼,𝑚_𝑔) and 𝑚^∼_𝑔≤max(𝑚₀,𝑚^∼,𝑚_𝑔), so that 𝑚=max(𝑚₀,𝑚^∼,𝑚_𝑔,𝑚^∼_𝑔), and 𝑉_ℎ𝑛(𝐷)⊆𝐷≠∅, 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}(𝐷_𝑓∪𝐷^∼_𝑓∪𝐷_𝑔)⊆𝐷_𝑓∪𝐷^∼_𝑓∪𝐷_𝑔≠∅ for all𝑛∈𝐍; where 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}(𝑢₁,…,𝑢_𝑚) is defined correspondingly to (2.5) for this case for all 𝑛∈𝐍;(2)𝜕(𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)+𝑔_𝑛(𝑋))/𝜕𝑋^𝑇|_𝑋^𝑐, 𝜕𝑔^∼_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋^𝑐 exist within a neighborhod of 𝑋^𝑐, which is a limit oscillatory solution of order at most 𝑚 of the vector controlled nominal equation, that is, the vector nominal uncontrolled via feedback of the nominal controller (including potentially nominal equilibrium points);(3) the inverse m-matrix (𝐼_𝑚−𝑀^𝑐(𝑋^𝑐))⁻¹ exists, where 𝑀^𝑐(𝑋^𝑐)=𝜕𝑉_ℎ(𝑋)/𝜕𝑋^𝑇|_𝑋^𝑐.Then 𝑋^𝑐𝑝=𝑋^𝑐+(𝐼_𝑚−𝑀^𝑐(𝑋^𝑐))⁻¹𝑉^∼_𝑔(𝑋^𝑐) is a linear estimate of limit oscillatory solutions of order at most 𝑚 (including potential equilibrium points as particular cases) of the vector controlled difference equation under the combined nominal and correction controllers from its corresponding counterpart under the nominal controller only. The estimate closes to true values as ‖𝑉^∼_𝑔(𝑋^𝑐)‖ is sufficiently small.
If rank(𝐼_𝑚−𝑀^𝑐(𝑋^𝑐),𝑉^∼_𝑔(𝑋^𝑐))=rank(𝐼_𝑚−𝑀^𝑐(𝑋^𝑐))<𝑚, then there are infinitely many first-order estimates 𝑋^𝑐𝑝 of limiting oscillatory solutions of the vector uncontrolled nominal equation of order at most 𝑚. If 𝑚−1≥rank(𝐼_𝑚−𝑀^𝑐(𝑋^𝑐),𝑉^∼_𝑓(𝑋^𝑐))>rank(𝐼_𝑚−𝑀^𝑐(𝑋^𝑐)), then there is no such an estimate.

The same linearization technique can be used to compare closely allocated equilibrium points of the same dimension for other pairs of the involved systems. In this way, the following results follow, respectively, for the nominal uncontrolled and controlled difference equations and for the uncontrolled nominal and controlled perturbed ones and, equivalently, for the associate pairs of vector systems as follows.

Theorem 2.7. Assume the following:(1)𝑚_𝑐∶=max(𝑚₀,𝑚_𝑔)=𝑚₀, 𝑉_𝑓𝑛(𝐷_𝑓)⊆𝐷_𝑓≠∅ and 𝑉_{𝑓𝑛+𝑔𝑛}(𝐷_𝑓∪𝐷_𝑔)⊆𝐷_𝑓∪𝐷_𝑔≠∅; forall𝑛∈𝐍, where 𝑉_{𝑓𝑛+𝑔𝑛}(𝑢₁,…,𝑢_𝑚𝑐) is defined correspondingly to (2.5) for this case for all 𝑛∈𝐍;(2)𝜕𝑓_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, 𝜕𝑔_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰ exist within a neighborhod of 𝑋⁰, which is a limit oscillatory solution of the vector uncontrolled nominal equation of order at most 𝑚₀ (including potentially nominal equilibrium points as particular cases);(3) the inverse 𝑚₀-matrix (𝐼_𝑚0−𝑀^𝑐(𝑋⁰))⁻¹ exists, where 𝑀^𝑐(𝑋⁰)=𝜕𝑉_𝑓(𝑋)/𝜕𝑋^𝑇|_𝑋⁰+𝜕𝑉_𝑔(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, and 𝐼_𝑚0  is the 𝑚₀-identity matrix of 𝐑^𝑚0.Then, 𝑋^𝑐=𝑋⁰+(𝐼_𝑚−𝑀^𝑐(𝑋⁰))⁻¹𝑉_𝑔(𝑋⁰) is a linear estimate of limit oscillatory solutions of order at most 𝑚₀ (including, as particular cases, potential equilibrium points) of the vector controlled difference equation from its nominal uncontrolled counterpart. The estimate closes to true values as ‖𝑉_𝑔(𝑋⁰)‖ is sufficiently small.
If rank(𝐼_𝑚0−𝑀^𝑐(𝑋⁰),𝑉_𝑔(𝑋⁰))=rank(𝐼_𝑚0−𝑀^𝑐(𝑋⁰))<𝑚₀, then there are infinitely many first-order estimates 𝑋^𝑐 of limiting oscillatory solutions of the vector controlled nominal equation of order at most 𝑚_𝑐=𝑚₀. If 𝑚₀−1≥rank(𝐼_𝑚0−𝑀^𝑐(𝑋⁰),𝑉_𝑔(𝑋⁰))>rank(𝐼_𝑚0−𝑀^𝑐(𝑋⁰)), then there is no such an estimate.

Theorem 2.8. Assume the following:(1)𝑚_𝑐𝑝∶=max(𝑚₀,𝑚^∼,𝑚_𝑔)=max(𝑚₀,𝑚^∼)=𝑚₀, 𝑉_𝑓𝑛(𝐷_𝑓∪𝐷^∼_𝑓)⊆𝐷_𝑓∪𝐷^∼_𝑓≠∅, and 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}(𝐷_𝑓∪𝐷^∼_𝑓∪𝐷_𝑔)⊆𝐷_𝑓∪𝐷^∼_𝑓∪𝐷_𝑔≠∅ for all 𝑛∈𝐍, where 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}(𝑢₁,…,𝑢_𝑚𝑐𝑝) is defined correspondingly to (2.5) for this case for all 𝑛∈𝐍;(2)𝜕𝑓_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, 𝜕𝑓^∼_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, 𝜕𝑔_𝑛(𝑋)/𝜕𝑋^𝑇|_𝑋⁰ exist within a neighborhod of 𝑋⁰, which is a limit oscillatory solution of the vector uncontrolled nominal equation of order at most 𝑚₀ (including potentially nominal equilibrium points as particular cases);(3) the inverse 𝑚₀-matrix (𝐼_𝑚0−𝑀^𝑐𝑝(𝑋⁰))⁻¹ exists, where 𝑀^𝑐𝑝(𝑋⁰)=𝜕𝑉_𝑓(𝑋)/𝜕𝑋^𝑇|_𝑋⁰+𝜕𝑉^∼_𝑓(𝑋)/𝜕𝑋^𝑇|_𝑋⁰+𝜕𝑉_𝑔(𝑋)/𝜕𝑋^𝑇|_𝑋⁰, and 𝐼_𝑚0 is the 𝑚₀-identity matrix of 𝐑^𝑚0.
Then, 𝑋^𝑐𝑝=𝑋⁰+(𝐼_𝑚−𝑀^𝑐𝑝(𝑋⁰))⁻¹𝑉^∼_𝑓+𝑔(𝑋⁰) is a linear estimate of limit oscillatory solutions of order at most 𝑚₀ (including, as particular cases, potential equilibrium points) of the vector controlled difference equation from its nominal uncontrolled counterpart. The estimate closes to true values as ‖𝑉^∼_𝑓+𝑔(𝑋⁰)‖ is sufficiently small.
If rank(𝐼_𝑚0−𝑀^𝑐𝑝(𝑋⁰),𝑉^∼_𝑓+𝑔(𝑋⁰))=rank(𝐼_𝑚0−𝑀^𝑐𝑝(𝑋⁰))<𝑚₀ then there are infinitely many first-order estimates 𝑋^𝑐𝑝 of limiting oscillatory solutions of the vector controlled nominal equation of order at most 𝑚_𝑐𝑝=𝑚₀. If 𝑚₀−1≥rank(𝐼_𝑚0−𝑀^𝑐𝑝(𝑋⁰),𝑉^∼_𝑓+𝑔(𝑋⁰))>rank(𝐼_𝑚0−𝑀^𝑐𝑝(𝑋⁰)), then there is no such an estimate.

3. Some Stability and Instability Properties

The following result holds concerning the stabilization via a feedback controller of an unstable uncontrolled equilibrium point. The controller consists, in general, of two parts, namely, (a) the nominal controller used to stabilize the uncontrolled difference equation; (b) the incremental controller used to stabilize the difference equation which includes perturbed parameters and/or perturbed dynamics. The stabilization process admits the double interpretation of the above section in terms of stabilization of either equilibrium points or that of oscillatory solutions. The equilibrium points can potentially vary under perturbations and the presence of feedback controllers.

Theorem 3.1. The following properties hold:(i)let 𝑥⁰and 𝑥^𝑐 be two equilibrium points of the nominal and nominal controlled difference equations with corresponding ones 𝑋⁰ and 𝑋^𝑐 in the associate vector equations. Assume that 𝑚=𝑚₀=𝑚_𝑐, 𝑋⁰∈cl𝑆∩𝐴₀ and 𝑋^𝑐∈cl𝑆∩𝐴_𝑐 with ∅≠𝑆⁰⊆𝑆⊆𝐴₀∩𝐴_𝑐, where 𝑆is an invariant subset of solutions of the associate vector equations for all 𝑉_𝑓𝑛 and 𝑉_{𝑓𝑛+𝑔𝑛}, that is, 𝑉_𝑓𝑛(𝑆)⊆𝑆and 𝑉_{𝑓𝑛+𝑔𝑛}(𝑆)⊆𝑆, for any controller in 𝐂 consisting in a nominal controller, with 𝐴₀∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)−𝑥⁰≥𝛼_𝑛𝑋−𝑋⁰,𝐴;∀𝑛∈𝐍_𝑐∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)+𝑔_𝑛(𝑋)−𝑥^𝑐≤𝛽^𝑐_𝑛𝑋−𝑋^𝑐;∀𝑛∈𝐍, (3.1) for some real nonnegative sequences {𝛼_𝑖}_𝑖∈𝐍and {𝛽^𝑐_𝑖}_𝑖∈𝐍. If {𝛼_𝑛}_𝑛∈𝐍 is unbounded, where 𝛼_𝑛=∏^𝑛_𝑖=1𝛼_𝑖, and {𝛽^𝑐_𝑛}_𝑛∈𝐍 is such that limsup_𝑛→∞𝛽^𝑐_𝑛<1, where 𝛽^𝑐_𝑛=∏^𝑛_𝑖=1𝛽^𝑐_𝑖, then 𝑋⁰ is unstable where 𝑋^𝑐 is locally asymptotically stable with respect to S;(ii) let 𝑥^0𝑝 and 𝑥^𝑐 be two equilibrium points of the uncontrolled perturbed and nominal controlled (via the nominal plus the incremental controllers) difference equations with corresponding ones 𝑋^0𝑝 and 𝑋^𝑐in the associate vector equations. If 𝑚=𝑚_0𝑝=max(𝑚₀,𝑚^∼)=𝑚_𝑐𝑝, 𝑋^0𝑝∈cl𝑆∩𝐴_0𝑝 and 𝑋^𝑐∈cl𝑆∩𝐴_𝑐 with ∅≠𝑆⁰⊆𝑆⊆𝐴_0𝑝∩𝐴_𝑐, where 𝑆 is an invariant subset of the solutions of the associate vector equations for all 𝑉_𝑓𝑛+^∼_𝑓𝑛 and 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}; that is, 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑆)⊆𝑆 and 𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛}(𝑆)⊆𝑆, with 𝐴_0𝑝∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)−𝑥^0𝑝≥𝛼_𝑝𝑛𝑋−𝑋^0𝑝,𝐴;∀𝑛∈𝐍_𝑐∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)+𝑔_𝑛(𝑋)−𝑥^𝑐≤𝛽^𝑐_𝑛𝑋−𝑋^𝑐;∀𝑛∈𝐍, (3.2) for some nonnegative sequences {𝛼_𝑝𝑖}_𝑖∈𝐍 and {𝛽^𝑐_𝑖}_𝑖∈𝐍. Thus, if {𝛼_𝑝𝑛}_𝑛∈𝐍 is unbounded, where 𝛼_𝑝𝑛=∏^𝑛_𝑖=1𝛼_𝑝𝑖, and {𝛽^𝑐_𝑛}_𝑛∈𝐍 is such that limsup_𝑛→∞𝛽^𝑐_𝑛<1, where 𝛽^𝑐_𝑛=∏^𝑛_𝑖=1𝛽^𝑐_𝑖, then 𝑋^0𝑝 is unstable where 𝑋^𝑐 is locally asymptotically stable with respect to S;(iii) let 𝑥^0𝑝 and 𝑥^𝑐𝑝 be two equilibrium points of the perturbed and perturbed controlled (via the nominal plus the incremental controllers) difference equations with corresponding ones 𝑋^0𝑝 and 𝑋^𝑐𝑝 in the associate vector equations. If 𝑚=𝑚_0𝑝=𝑚_𝑐𝑝, 𝑋^0𝑝∈cl𝑆∩𝐴_0𝑝, and 𝑋^𝑐𝑝∈cl𝑆∩𝐴_𝑐𝑝 with ∅≠𝑆⁰⊆𝑆⊆𝐴_0𝑝∩𝐴_𝑐𝑝, where 𝑆 is an invariant subset of the solutions of the associate vector equations for all 𝑉_𝑓𝑛+^∼_𝑓𝑛 and 𝑉_ℎ𝑛, that is, 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑆)⊆𝑆 and 𝑉_ℎ𝑛(𝑆)⊆𝑆, for some nonnegative real sequences {𝛼_𝑖}_𝑖∈𝐍 and {𝛽^𝑐_𝑝𝑖}_𝑖∈𝐍, and 𝐴_𝑐𝑝∶=𝑋∈𝑅^𝑚∶ℎ_𝑛(𝑋)−𝑥^𝑐𝑝≤𝛽^𝑐_𝑝𝑛𝑋−𝑋^𝑐𝑝;∀𝑛∈𝐍. (3.3) Thus, if {𝛼_𝑛}_𝑛∈𝐍 is unbounded, where 𝛼_𝑛=∏^𝑛_𝑖=1𝛼_𝑖, and {𝛽^𝑐_𝑝𝑛}_𝑛∈𝐍 is such that limsup_𝑛→∞𝛽^𝑐_𝑝𝑛<1, where 𝛽^𝑐_𝑝𝑛=∏^𝑛_𝑖=1𝛽^𝑐_𝑝𝑖, then 𝑋^0𝑝 is unstable where 𝑋^𝑐𝑝 is locally asymptotically stable with respect to 𝑆.

The above result is extendable to stabilization of unstable oscillatory solutions in the light of the former discussions in Section 2. Explicit conditions for the fulfilment of Theorem 3.1(iii), which imply the local asymptotic stabilization within an invariant set around the equilibrium points of the unstable perturbed uncontrolled system, are given in the subsequent result. The stabilization mechanism is achieved by synthesizing a controller consisting of combined nominal controller with an incremental controller. The nominal controller stabilized the nominal difference equation in the absence of perturbations, while the incremental one completes the stabilization for the whole uncontrolled difference equation.

Theorem 3.2. Assume that 𝑚=𝑚_0𝑝=𝑚_𝑐𝑝 with 𝑋^0𝑝∈cl𝑆∩𝐴₀ and 𝑋^𝑐𝑝∈cl𝑆∩𝐴_𝑐𝑝 being unique equilibrium points in cl𝑆∩𝐴₀, respectively in cl𝑆∩𝐴_𝑐𝑝, where 𝑆⊆𝐴_0𝑝∩𝐴_𝑐𝑝 is invariant under all 𝑉_𝑓𝑛+^∼_𝑓𝑛 and 𝑉_ℎ𝑛 for a class of controllers 𝐂, that is, 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑆)⊆𝑆 and 𝑉_ℎ𝑛(𝑆)⊆𝑆 for any combined nominal plus incremental controller in the class 𝐂 for all 𝑛∈𝐍. Define Δ𝑥^0𝑝∶=𝑥^𝑐𝑝−𝑥^0𝑝 and Δ𝑋^0𝑝∶=𝑋^𝑐𝑝−𝑋^0𝑝 being sufficiently close to zero to satisfy Δ𝑋^0𝑝≤𝛽^𝑐_𝑝𝑛⁻¹𝛼_𝑝𝑛−𝛽^𝑐_𝑝𝑛𝑋−𝑋^0𝑝,∀𝑋∈cl𝑆,∀𝑛∈𝐍. (3.4) Assume also that the nominal and incremental controllers are 𝑔_𝑛𝑋_𝑛−1=𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛};𝑔^∼_𝑛𝑋_𝑛−1=𝜆^∼_𝑛𝑋𝑓_𝑛−1^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛;∀𝑛∈𝐍, (3.5) with their corresponding gain sequences {𝜆_𝑛}_𝑛∈𝐍 and {𝜆^∼_𝑛}_𝑛∈𝐍 being chosen to satisfy the constraints: sign𝜆_𝑛𝑋_𝑛−1=−sign𝑓_𝑛𝑋_𝑛−1−𝑥^0𝑝𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛};∀𝑛∈𝐍,sign𝜆^∼_𝑛𝑋_𝑛−1=−sign𝑓^∼_𝑛𝑋_𝑛−1−𝑥𝑓^0𝑝∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛𝜆;∀𝑛∈𝐍,_𝑛𝑋_𝑛−1,𝜆^∼_𝑛𝑋_𝑛−1≤𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^𝑐𝑝𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛},𝑓^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛;∀𝑛∈𝐍, (3.6)(3.7)(3.8) for some existing nonnegative integer sequences {𝜎_𝑛}_𝑛∈𝐍0, {𝜎_𝑛}_𝑛∈𝐍0 chosen such that ‖(𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛}),𝑓^∼_𝑛−𝜎_𝑛(𝑋_𝑛−1−𝜎_𝑛))‖≠0 for all 𝑛∈𝐍₀ subject to liminf_𝑛→∞‖(𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛}),𝑓^∼_𝑛−𝜎_𝑛(𝑋_𝑛−1−𝜎_𝑛))‖≥0since equality to zero holds for all nonnegative sequences {𝜎_𝑛}_𝑛∈𝐍0, {𝜎_𝑛}_𝑛∈𝐍0 if and only if 𝑋^𝑐𝑝=0.
Then, the corresponding equilibrium point of the perturbed uncontrolled associate vector system 𝑋^0𝑝 is unstable, while that of the perturbed controlled system 𝑋^𝑐𝑝 is asymptotically stable. The properties hold for the corresponding perturbed and perturbed controlled difference equations of equilibrium points 𝑥^0𝑝 and 𝑥^𝑐𝑝, respectively.

Proof. Conditions for the following chain of inequalities to hold are given: ℎ_𝑛𝑋_𝑛−1−𝑥^𝑐𝑝=𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1+𝑔_𝑛𝑋_𝑛−1+𝑔^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝−Δ𝑥^0𝑝≤𝛽^𝑐_𝑝𝑛𝑋_𝑛−1−𝑋^𝑐≤𝛼_𝑝𝑛𝑋_𝑛−1−𝑋^0𝑝≤𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝;∀𝑛∈𝐍 (3.9) within 𝑆. The following chained inequalities guarantee that (3.9) holds in 𝑆: 𝛼_𝑝𝑛𝑋−𝑋^0𝑝−𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}+𝜆^∼_𝑛𝑋𝑓_𝑛−1^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛−Δ𝑥^0𝑝≤𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝+𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}+𝜆^∼_𝑛𝑋𝑓_𝑛−1^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛−Δ𝑥^0𝑝=𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝−Δ𝑥^0𝑝−𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}+𝜆^∼_𝑛𝑋𝑓_𝑛−1^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛≤𝛽^𝑐_𝑝𝑛𝑋−𝑋^0𝑝−Δ𝑋^0𝑝≤𝛼_𝑝𝑛𝑋−𝑋^0𝑝𝑓;∀𝑛∈𝐍,0≤_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝−Δ𝑥^0𝑝−𝛽^𝑐_𝑝𝑛𝑋−𝑋^0𝑝−Δ𝑋^0𝑝≤𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}+𝜆^∼_𝑛𝑋𝑓_𝑛−1^∼_𝑛−𝜎_𝑛𝑋_𝑛−1−𝜎_𝑛;∀𝑛∈𝐍. (3.10) Then, the nominal and incremental controller gains are chosen to satisfy (3.6)–(3.8) for existing nonnegative real sequences {𝜎_𝑛}_𝑛∈𝐍0, {𝜎_𝑛}_𝑛∈𝐍0 such that ‖(𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛}),𝑓^∼_𝑛−𝜎_𝑛(𝑋_𝑛−1−𝜎_𝑛))‖≠0 for all 𝑛∈𝐍₀ subject to liminf_𝑛→∞‖(𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛}),𝑓^∼_𝑛−𝜎_𝑛(𝑋_𝑛−1−𝜎_𝑛))‖≥0since equality to zero holds if and only if 𝑋^𝑐𝑝=0. The reminder of the proof now follows since from (3.9) one gets simultaneously within a nonempty invariant set 𝑆⊆𝐴_0𝑝∩𝐴_𝑐𝑝: ℎ_𝑛𝑋_𝑛−1−𝑥^𝑐𝑝≤𝛽^𝑐_𝑝𝑛𝑋_𝑛−1−𝑋^𝑐𝑝;∀𝑋_𝑖∈𝑆,∀𝑖∈𝐍₀𝛼_𝑝𝑛,𝑋_𝑛−1−𝑋^0𝑝≤𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝;∀𝑋_𝑖∈𝑆,∀𝑖∈𝐍₀. (3.11)(3.12) Then, one gets from (3.12): ℎ_𝑛(𝑋)−𝑥^𝑐𝑝=ℎ_𝑛𝑉_ℎ𝑛−1(𝑋)−𝑋^𝑐𝑝=ℎ_𝑛𝐺^ℎ_𝑛−1(𝑋)−𝑥^𝑐𝑝≤𝛽^𝑐_𝑝𝑛𝑋−𝑋^𝑐𝑝≤^𝑛_𝑖=1max_{()𝑖≥𝑗∈𝐍≥𝑖−𝑚−1}𝛽^𝑐_𝑝𝑗×𝑋−𝑋^𝑐𝑝≤𝛽^𝑐𝑛𝑋−𝑋^𝑐𝑝;∀𝑋∈cl𝑆,∀𝑛∈𝐍, (3.13) with the real sequence {𝛽^𝑐𝑛}_𝑛∈𝐍0 of elements satisfying 𝛽^𝑐𝑛∈[0,1), forall𝑛∈𝐍 so that one deduces by taking ℓ_∞-norms for the 𝑚-tuples 𝐺^ℎ_(⋅)(𝑋) (see Remark 2.3) that: 𝐺^ℎ_𝑛−1(𝑋)−𝑋^𝑐𝑝_∞=max_{()𝑛≥𝑗∈𝐍≥𝑛−𝑚−1}ℎ_𝑗𝐺^ℎ_𝑗−1(𝑋)−𝑥^𝑐𝑝≤𝛽^𝑐𝑛𝑋−𝑋^𝑐𝑝_∞;∀𝑛∈𝐍. (3.14) Since 𝛽^𝑐𝑛∈(0,1) for all 𝑛∈𝐍, then lim_𝑛→∞(𝐺^ℎ_𝑛−1(𝑋)−𝑋^𝑐𝑝)=0 for all 𝑋∈𝑆. Since {𝐺^ℎ_𝑛(𝑋)}_𝑛∈𝐍 is a sequence of contraction self-mappings from 𝐑^𝑚∣𝑆 to 𝑆, and 𝐑^𝑚 is a complete metric space endowed with the given norm-induced metric, then the equilibrium point 𝑋^𝑐𝑝 on the vector function equation associated to the controlled difference equation is locally asymptotically stable with respect to 𝑆, and it is also the unique fixed point in cl𝑆, [8]. Then, the equilibrium point 𝑋^𝑐𝑝 on the controlled difference equation is locally asymptotically stable with respect to 𝑆, and the equilibrium point 𝑥^𝑐𝑝 of the corresponding difference equation is also locally stable. On the other hand, it follows from (3.12) that 𝛼_𝑝𝑛−1𝑋−𝑋^0𝑝_∞=𝛼_𝑝𝑛𝐺^𝑓+∼𝑓_𝑛−1(𝑋)−𝑋^0𝑝_∞≤𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)−𝑥^0𝑝=𝑓_𝑛𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑋)+𝑓^∼_𝑛𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑋)−𝑥^0𝑝≤𝐺^𝑓+∼𝑓_𝑛(𝑋)−𝑋^0𝑝_∞;∀𝑋∈cl𝑆,∀𝑛∈𝐍. (3.15) Since {𝛼_𝑝𝑛}_𝑛∈𝐍 is unbounded with 𝛼_𝑝𝑛∶=∏^𝑛_𝑖=1𝛼_𝑝𝑖, then {𝐺^𝑓+∼𝑓_𝑛(𝑋)}_𝑛∈𝐍 is a sequence of expanding self-mappings from 𝐑^𝑚∣𝑆 to 𝑆 so that the equilibrium point 𝑋^0𝑝 of the vector function equation associated to the perturbed uncontrolled difference equation is locally unstable with respect to 𝑆. Then, the corresponding equilibrium point of the difference equation 𝑥^0𝑝 is unstable.

The next result is concerned with the local asymptotic stabilization around equilibrium points within a certain invariant set of the unstable uncontrolled perturbed difference equation through a single controller provided that the perturbation dynamics satisfies some small-type constraints.

Theorem 3.3. Assume the following:(1)𝑚=𝑚_0𝑝=𝑚_𝑐 with 𝑋^0𝑝∈cl𝑆∩𝐴_0𝑝, and 𝑋^𝑐∈cl𝑆∩𝐴_𝑐 being unique equilibrium points in cl𝑆∩𝐴_0𝑝, respectively in cl𝑆∩𝐴_𝑐, where 𝑆⊆𝐴_0𝑝∩𝐴_𝑐 is invariant under all 𝑉_𝑓𝑛+^∼_𝑓𝑛, and 𝑉_ℎ𝑛≡𝑉_𝑓𝑛+^∼_{𝑓𝑛+𝑔𝑛} (since 𝑔^∼_𝑛≡0) for any controller in the class 𝐂 consisting of a nominal controller, that is, 𝑉_𝑓𝑛+^∼_𝑓𝑛(𝑆)⊆𝑆 and 𝑉_ℎ𝑛(𝑆)⊆𝑆 with the incremental controller being identically zero for all 𝑛∈𝐍, provided that the sets are nonempty. Assume also that the sets 𝐴_0𝑝 and 𝐴_𝑐 defined in (3.2) are redefined as 𝐴_0𝑝∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)−𝑥^0𝑝≥𝛼_𝑝𝑛1−𝛼^∼_𝑝𝑛𝑋−𝑋^0𝑝,𝐴;∀𝑛∈𝐍_𝑐∶=𝑋∈𝑅^𝑚∶𝑓_𝑛(𝑋)+𝑓^∼_𝑛(𝑋)+𝑔_𝑛(𝑋)−𝑥^𝑐≤𝛽^𝑐_𝑛1+𝛽^∼𝑐_𝑛𝑋−𝑋^𝑐;∀𝑛∈𝐍, (3.16) for nonnegative real sequences {𝛼_𝑝𝑛(1−𝛼^∼_𝑝𝑛)}_𝑛∈𝐍 and {𝛽^𝑐_𝑛(1+𝛽^∼𝑐_𝑛)}_𝑛∈𝐍 being defined for some nonnegative real sequences {𝛼^∼_𝑝𝑛}_𝑛∈𝐍 and {𝛽^∼𝑐_𝑛}_𝑛∈𝐍, subject to 𝛼^∼_𝑝𝑛≤1 and 𝛽^∼𝑐_𝑛<𝛽_𝑛⁻¹−1, where {𝛼_𝑝𝑛}_𝑛∈𝐍 is unbounded of elements redefined as 𝛼_𝑝𝑛∶=∏^𝑛_𝑖=1[𝛼_𝑝𝑖(1−𝛼^∼_𝑝𝑖)], and {𝛽^𝑐_𝑛}_𝑛∈𝐍 of elements being redefined as 𝛽^𝑐_𝑛∶=∏^𝑛_𝑖=1[𝛽^𝑐_𝑖(1+𝛽^∼𝑐_𝑖)] being such that limsup_𝑛→∞𝛽^𝑐_𝑛<1;(2) the perturbed sequence {𝑓^∼_𝑛(𝑋_𝑛−1)}_𝑛∈𝐍 satisfies the constraints: 𝑓^∼_𝑛𝑋_𝑛−1−𝑎_𝑛≤𝛽𝛽^𝑐_𝑛^∼𝑐_𝑛𝑋_𝑛−1−𝑋^𝑐;𝑓^∼_𝑛𝑋_𝑛−1−𝑏_𝑛≤𝛼^∼_𝑝𝑛𝛼_𝑝𝑛𝑋_𝑛−1−𝑋^0𝑝, (3.17) for all 𝑛∈𝐍 within 𝑆 for some real sequences {𝑎_𝑛}_𝑛∈𝐍and {𝑏_𝑛}_𝑛∈𝐍 and for some nonnegative real sequences {𝛼^∼_𝑝𝑛}_𝑛∈𝐍0 and {𝛽^∼𝑐_𝑛}_𝑛∈𝐍0, subject to 𝛼^∼_𝑝𝑛≤1 and 𝛽^∼𝑐_𝑛<𝛽_𝑛⁻¹−1;(3) the stabilizing incremental controller is identically zero, while the nominal controller is 𝑔_𝑛(𝑋_𝑛−1)=𝜆_𝑛(𝑋_𝑛−1)𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛}) subject to sign𝜆_𝑛𝑋_𝑛−1=−sign𝑓_𝑛𝑋_𝑛−1−𝑥^0𝑝𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}𝜆;∀𝑛∈𝐍,_𝑛𝑋_𝑛−1≤𝑓_𝑛𝑋_𝑛−1−𝑥^𝑐+𝑎_𝑛𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛};∀𝑛∈𝐍, (3.18) within 𝑆 if 𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛})≠0 and 𝜆_𝑛(𝑋_𝑛−1)=0 if 𝑓_{𝑛−𝜎𝑛}(𝑋_{𝑛−1−𝜎𝑛})=0 for all 𝑛∈𝐍, for some existing non-negative integer sequence {𝜎_𝑛}_𝑛∈𝐍0.
Then, the corresponding equilibrium point of the perturbed uncontrolled associate vector system 𝑋^0𝑝 is unstable, while that of the controlled system 𝑋^𝑐 under the nominal controller is asymptotically stable. The stability properties also hold for the corresponding perturbed and perturbed controlled difference equations of equilibrium points 𝑥^0𝑝 and 𝑥^𝑐, respectively.

Proof. Now the errors of the uncontrolled perturbed and the controlled nominal equilibrium points under consideration are Δ𝑥^0𝑝∶=𝑥^𝑐−𝑥^0𝑝 and Δ𝑋^0𝑝∶=𝑋^𝑐−𝑋^0𝑝. Note that 𝑔^∼_𝑛(𝑋_𝑛−1)≡0 so that (3.7) is omitted, while (3.6) and (3.8) are replaced by (3.18). Also, the inequalities (3.11)-(3.12) are replaced by 𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1+𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}−𝑥^𝑐≤𝑓_𝑛𝑋_𝑛−1+𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}−𝑥^𝑐+𝑎_𝑛+𝑓^∼_𝑛𝑋_𝑛−1−𝑎_𝑛≤𝛽^𝑐_𝑛1+𝛽^∼𝑐_𝑛𝑋_𝑛−1−𝑋^𝑐;∀𝑋_𝑛∈𝑆,∀𝑛∈𝐍₀,𝛼_𝑝𝑛1−𝛼^∼_𝑝𝑛𝑋_𝑛−1−𝑋^0𝑝≤𝑓_𝑛𝑋_𝑛−1−𝑥^0𝑝+𝑏_𝑛−𝑓^∼_𝑛𝑋_𝑛−1−𝑏_𝑛≤𝑓_𝑛𝑋_𝑛−1+𝑓^∼_𝑛𝑋_𝑛−1−𝑥^0𝑝;∀𝑋_𝑛∈𝑆,∀𝑛∈𝐍₀, (3.19)(3.20) for all 𝑛∈𝐍₀, provided that 𝑓_𝑛𝑋_𝑛−1+𝜆_𝑛𝑋_𝑛−1𝑓_{𝑛−𝜎𝑛}𝑋_{𝑛−1−𝜎𝑛}−𝑥^𝑐+𝑎_𝑛≤𝛽^𝑐_𝑛𝑋_𝑛−1−𝑋^𝑐;∀𝑋_𝑛∈𝑆,∀𝑛∈𝐍₀,𝛼_𝑝𝑛𝑋_𝑛−1−𝑋^0𝑝≤𝑓_𝑛𝑋_𝑛−1−𝑥^0𝑝+𝑏_𝑛;∀𝑋_𝑛∈𝑆,∀𝑛∈𝐍₀, (3.21) for some existing nonnegative sequences {𝛼_𝑝𝑛(1−𝛼^∼_𝑝𝑛)}_𝑛∈𝐍 and {𝛽^𝑐_𝑛(1+𝛽^∼𝑐_𝑛)}_𝑛∈𝐍. Thus, 𝑋^0𝑝 is unstable, while 𝑋^𝑐is locally asymptotically stable with respect to 𝑆. Those properties also hold by construction of 𝑋^0𝑝 and 𝑋^𝑐 for the corresponding perturbed and perturbed controlled difference equations of equilibrium points 𝑥^0𝑝 and 𝑥^𝑐, respectively.

Remark 3.4. Note that in the proof of the results of this section, estimates can replace to the true equilibrium points if they are, potentially distinct, but sufficiently close to each other by using the results of Section 2 provided that the needed assumptions of the various function smoothness hold. Furthermore, the equilibrium points under analysis in the various given results could be replaced with the estimates of errors related to the nominal equilibrium if such errors are sufficiently small in terms of smallness of error norms. For instance, take the estimation error of the equilibrium points of the feedback associate vector equation via the nominal controller compared to its uncontrolled perturbed counterpart: Δ∶=𝑋^𝑐−𝑋⁰+𝐼_𝑚−𝑀^𝑐𝑋⁰⁻¹𝑉^∼_𝑓+𝑔𝑋⁰, (3.22) for all 𝑋∈𝑆 calculated from Theorem 2.8. Thus, the last term of (3.19) with 𝛽^∼𝑐_𝑛=0 for all 𝑛∈𝐍₀ possesses the lower-bounds given below: 𝛽^𝑐_𝑛𝑋−𝑋^𝑐=𝛽^𝑐_𝑛𝑋−𝑋^𝑐−Δ≥𝛽^𝑐_𝑛𝑋−𝑋^𝑐−𝛽^𝑐_𝑛‖Δ‖≥𝛽^𝑐_𝑛1−𝜀^𝑐_𝑛𝑋−𝑋^𝑐, (3.23) provided that the equilibrium point estimation error is of sufficiently small size fulfilling ‖Δ‖≤𝜀^𝑐_𝑛‖𝑋−𝑋^𝑐‖ for some sequence {𝜀^𝑐_𝑛}_𝑛∈𝐍0 satisfying 0≤𝜀^𝑐_𝑛≤1 for all 𝑛∈𝐍₀. Thus, the last term of (3.19) could be replaced by (3.23), and the theorem and its proof could be reformulated based on estimates when having a sufficiently small estimation error between the equilibrium point of the uncontrolled difference equation and that of the current controlled difference equation, that is, that subject to parametrical perturbation with or without unmodeled dynamics.

4. Example

Consider the following difference equation:𝑥_𝑛+1=ℎ_𝑛𝑥_𝑛=𝑓_𝑛𝑥_𝑛+𝑓^∼_𝑛𝑥_𝑛+𝑔_𝑛𝑥_𝑛=𝛿_𝑛+𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛+𝑔_𝑛𝑥_𝑛, (4.1) where 𝛿_𝑛≥1 for all 𝑛∈𝐍, 𝑓_𝑛(𝑥_𝑛)=𝛿_𝑛𝑥_𝑛𝑒^−𝑥𝑛 is the uncontrolled nominal dynamics, 𝑓^∼_𝑛(𝑥_𝑛)=𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛 is the unmodeled uncontrolled dynamics, and 𝑔_𝑛(𝑥_𝑛) is a controller with the objective of stabilizing (1.1) for a certain size of the unmodeled dynamics added to the nominal uncontrolled one. Equation (4.1) extends with the incorporation of uncertain dynamics and nominal and incremental controls the example given in [12] for the case of fully modelled dynamics. Note that an equilibrium point of the uncontrolled nominal and perturbed difference equations is 𝑥=0. If 𝛿_𝑛→𝛿>1 as 𝑛→∞, then 𝑥=ln𝛿is also an equilibrium point of such an equation.

The particular 𝑥_𝑛+1=ℎ_𝑛(𝑥_𝑛)=𝑓_𝑛(𝑥_𝑛) is the nominal uncontrolled part of (4.1). If 𝛿_𝑛≥1 for all 𝑛∈𝐍, 𝑆≡𝐑—(the set of negative real numbers) and 𝐴₀∶={𝑥∈𝐑∶|𝑓_𝑛(𝑥)|≥𝛿_𝑛|𝑥|}, then 𝑆⊂𝐴₀ is invariant.

The equilibrium point 𝑥=0  of the uncontrolled nominal difference equations is unstable with respect to 𝑆;  𝑥₀=0⇒𝑥_𝑛=0; for all𝑛∈𝐍, and if 𝑥₀>0 and finite, then{𝑥_𝑛}_𝑛∈𝐍 is nonnegative and uniformly bounded. To see this property, proceed with a contradiction argument as follows. Assume that {𝑥_𝑛}_𝑛∈𝐍 is unbounded, then if {𝑥_𝑛}_𝑛∈𝐍 is unbounded, then there exists a subsequence {𝑥_𝑛}_𝑁𝑛∈ of it which diverges, that is, 𝑥_𝑛→+∞ as 𝑛(∈𝑁)→∞ with 𝑁⊂𝐍 is numerable of infinite cardinal. As a result, there is a monotone increasing real sequence {𝑀_𝑛}_𝑁𝑛∈ such that 𝑀_𝑛+1≥𝑥_𝑛>𝑀_𝑛 so that 𝑀_𝑛+1≤𝑥_𝑛+1≤𝛿_𝑛𝑥_𝑛𝑒^−𝑀𝑛≤𝛿_𝑛𝑀_𝑛+1𝑒^−𝑀𝑛 for all 𝑛∈𝑁 so that 1≤liminf_{𝑛(∈𝑁)→∞}𝛿_𝑛𝑒^−𝑀𝑛=0, which is clearly a contradiction to {𝑥_𝑛}_𝑛∈𝐍 being unbounded by assuming that {𝛿_𝑛}_𝑁𝑛∈ is not exponentially unbounded. Then, {𝑥_𝑛}_𝑛∈𝐍 is bounded if 𝑥₀≥0 and 𝛿_𝑛≥1 for all 𝑛∈𝐍. The above contradiction argument does not require specifically 𝛿_𝑛≥1, but only 𝛿_𝑛>0 for all 𝑛∈𝐍. Thus, {𝑥_𝑛}_𝑛∈𝐍 is bounded if 𝑥₀≥0 and 𝛿_𝑛≥0 for all 𝑛∈𝐍, and then the equilibrium point 𝑥=0 of the uncontrolled nominal difference equations is locally stable with respect to the complement 𝑆 of 𝑆 in 𝐑 which is also an invariant set from Theorem 3.3. As a result, the zero equilibrium point is unstable with respect to 𝑆 if 𝛿_𝑛≥1 for all 𝑛∈𝐍 and globally stable with respect to 𝑆 if 𝛿_𝑛>0 for all 𝑛∈𝐍. A combined result is 𝑥=0 is a semistable equilibrium point if 𝛿_𝑛≥1.

One now considers the synthesis of a stabilizing controller 𝑔_𝑛(𝑥_𝑛)=𝜆_𝑛𝛿_𝑛𝑥_𝑛𝑒^−𝑥𝑛 for the nominal dynamics and at the same time for a certain tolerance to certain amount of unmodeled dynamics according to the controller of Theorem 3.3. A more sophisticated controller according to Theorem 3.2 could also be synthesized. Note that an equilibrium point of the whole controlled difference equation 𝑥_𝑛+1=ℎ_𝑛(𝑥_𝑛) is still 𝑥=0 as for the uncontrolled nominal and perturbed ones. Equation (4.1) becomes𝑥_𝑛+1=ℎ_𝑛𝑥_𝑛=1+𝜆_𝑛𝛿_𝑛+𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛;𝑛∈𝐍₀. (4.2) It is required from Theorem 3.3, (3.18) with 𝛿_𝑛=𝛼_𝑝𝑛, 𝜎_𝑛=0, that 𝜆_𝑛<0 if 𝑥₀>0 and 𝜆_𝑛>0 if 𝑥_𝑛<0 and for some real sequence {𝑎_𝑛}_𝑛∈𝐍0, and a gain sequence satisfying:𝜆_𝑛=𝛿_𝑛𝑥_𝑛+𝑎_𝑛𝑒^𝑥𝑛𝛿_𝑛𝑥_𝑛=𝑎1+^𝑛𝑒^𝑥𝑛𝛿_𝑛𝑥_𝑛if𝑥₀≥0with𝑎_𝑛<0,𝑎_𝑛≥𝛿_𝑛𝑥_𝑛𝑒^−𝑥𝑛;𝑎1−^𝑛𝛿_𝑛𝑥_𝑛𝑒^|𝑥𝑛|if𝑥_𝑛<0with𝑎_𝑛≥𝛿_𝑛𝑥_𝑛𝑒^|𝑥𝑛|;∀𝑛∈𝐍₀. (4.3) Thus,

(a)if 𝑥₀≥0, then |1+𝑎_𝑛𝑒^𝑥𝑛/𝛿_𝑛𝑥_𝑛|=|𝑎_𝑛𝑒^𝑥𝑛/𝛿_𝑛𝑥_𝑛|−1 so that one gets from (4.2): 𝑥_𝑛+1=𝑎_𝑛+𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛<𝑥_𝑛𝑒^−𝑥𝑛<𝑥_𝑛since𝑥_𝑛>0;∀𝑛∈𝐍₀, (4.4) provided that the controller gain sequence and unmodeled dynamics contribution sequence satisfy: 𝜆_𝑛𝑎=−1−^𝑛𝑒^𝑥𝑛𝛿_𝑛𝑥_𝑛,𝑎_𝑛≥𝛿_𝑛𝑥_𝑛𝑒^−𝑥𝑛;∀𝑛∈𝐍₀,_∼𝛿_𝑛∈𝛿_1𝑛,𝛿_2𝑛≡−𝑎_𝑛𝑒^𝑥𝑛𝑥_𝑛𝑎,1−^𝑛𝑒^𝑥𝑛𝑥_𝑛⊆−𝛿_𝑛,1−𝛿_𝑛;∀𝑛∈𝐍₀, (4.5)(4.6) where 𝑓^∼_𝑛(𝑥_𝑛)=𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛 for all 𝑛∈𝐍₀ is subject to the constraints (4.6) so as to define the sets (3.16) in Theorem 3.3 according to such a disturbance. Note that the constraint (4.5) implies that the unmodeled dynamics contribution is “small enough” to satisfy (3.17) in Theorem 3.3;(b) if 𝑥₀<0 then for any 𝑛∈𝐍₀ such that 𝑥_𝑛<0, and |1+𝑎_𝑛𝑒^𝑥𝑛/𝛿_𝑛𝑥_𝑛|=|𝑎_𝑛𝑒^𝑥𝑛/𝛿_𝑛𝑥_𝑛|−1 so that one gets from (4.2) that 𝑥_𝑛+1=−𝑎_𝑛+𝛿^∼_𝑛𝑥_𝑛𝑒^|𝑥𝑛|<−𝑥_𝑛≤−𝛽_𝑛𝑥_𝑛;∀𝑛∈𝐍₀, (4.7) for any sequence {𝛽_𝑛}_𝑛∈𝐍0 with any 𝛽_𝑛<1 for all 𝑛∈𝐍₀ provided that the controller gain sequence and unmodeled dynamics contribution sequence satisfy

𝜆_𝑛=𝑎1−^𝑛𝛿_𝑛𝑥_𝑛𝑒^|𝑥𝑛|,𝑥_𝑛≥𝑎_𝑛>0;∀𝑛∈𝐍₀,_∼𝛿_𝑛≤𝑎_𝑛𝑥_𝑛−1𝑒^{|−𝑥𝑛|};∀𝑛∈𝐍₀. (4.8)(4.9) Note that for both cases (a)-(b), |(1+𝜆_𝑛)𝛿_𝑛+𝛿^∼_𝑛|<1 for all 𝑛∈𝐍₀, and {𝑥_𝑛}_𝑛∈𝐍 is bounded (for any |𝑥₀| finite) and monotone decreasing so that it converges asymptotically to the equilibrium point. The stabilization of the zero equilibrium has been achieved irrespective of the values of the elements of the sequence {𝛿_𝑛}_𝑛∈𝐍0. Theorem 3.3 is fulfilled also for the sets (3.16) with 𝛼^∼_𝑝𝑛=𝛽^∼𝑐_𝑛=0, 𝛼_𝑝𝑛=𝛿_𝑛≥1 and 𝛽^𝑐_𝑛<1 for all 𝑛∈𝐍₀. The equilibrium point is asymptotically stable related to both the invariant set 𝑆 and its complement in 𝐑 so that it is also globally asymptotically Lyapunov stable. The discussion is summarized as follows.

Theorem 4.1. The equilibrium point 𝑥=0 of the difference equation of Section 4 is globally asymptotically stable around the equilibrium point 𝑥=0 (so that it is asymptotically stable related to both the invariant set 𝑆 and its complement in 𝐑) under the feedback controller 𝑔_𝑛(𝑥_𝑛)=𝜆_𝑛𝛿_𝑛𝑥_𝑛𝑒^−𝑥𝑛 for all 𝑛∈𝐍₀ given by the gain sequence of elements satisfying (4.5) if 𝑥₀≥0 (implying that 𝑥_𝑛≥0 for all 𝑛∈𝐍) and (4.8) if 𝑥_𝑛<0 for the current 𝑛∈𝐍₀ for any structured unmodeled dynamics 𝑓^∼_𝑛(𝑥_𝑛)=𝛿^∼_𝑛𝑥_𝑛𝑒^−𝑥𝑛, subject to (4.6) if 𝑥₀≥0, and to (4.9) if 𝑥₀<0 provided that 𝑥_𝑛<0 for the current 𝑛∈𝐍₀.

Acknowledgments

The author thanks the Spanish Ministry of Education for its support to this paper through Grant no. DPI2009-07197. He is also grateful to the Basque Government for its support through Grant no. IT378-10. He is also grateful to the referees for their useful comments.