# About the Stabilization of a Nonlinear Perturbed Difference Equation

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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.


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## Body

## 1. Introduction

In this paper, the following nonautonomous difference equation is investigated:(1.1)xn=hn(xn-1,…,xn-m)=fn(xn-1,…,xn-m0)+f̃n(xn-1,…,xn-m̃)+gn(xn-1,…,xn-mg)+g̃n(xn-1,…,xn-m̃g)=xn0+x̃n+xnc+x̃nc;n∈N,
of order m:=max(m0,m̃,mg,m̃g)≥1 and initial conditions x1-m,…,x0, where the four terms of the second identity are pair-wise identical in the same order as written, in which hn:D⊂Rm→R, fn:Df⊂Rm0→R, f̃n:Df̃⊂Rm̃→R, gn:Dg⊂Rmg→R, and g̃n:Dg̃⊂Rm̃g→R; D is a nonempty subset of the union of the sets Df, Df̃, Dg, and Dg̃. The four pair-wise identical terms of the last identity have the following interpretations:(i)
xn0 is the nominal value of the uncontrolled nominal solution xn at the nth sample in the absence of perturbations and controls;(ii)
x̃n is the perturbed uncontrolled solution which can be generated for perturbed parameterizations (then m̃≤m0) and possibly contributed by unmodeled dynamics (then m̃>m0);(iii)
xnc 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)
x̃nc 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 f̃n+gn+g̃n≡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 is(2.1)Vh(u1,…,um)=(h(u1,…,um),u1,…,um-1)=Vf(u1,…,um)+Vf̃(u1,…,um)+Vg(u1,…,um)+Vg̃(u1,…,um)=(f(u1,…,um0)+f̃(u1,…,um̃)+g(u1,…,umg)+g̃(u1,…,um̃g),u1,…,um-1),
where(2.2)Vf(u1,…,um):=(f(u1,…,um0),u1,…,um-1),Vf̃(u1,…,um):=(f̃(u1,…,um̃),u1,…,um-1),Vg(u1,…,um):=(g(u1,…,umg),u1,…,um-1),Vg̃(u1,…,um)=(g̃(u1,…,umg̃),u1,…,um-1).
In particular, Vh(u1)=(h(u1),u1) if m=1, one has the following particular case of  (2.2)(2.3)Vf(u1):=(f(u1));Vf̃(u1):=(f̃(u1));Vg(u1):=(g(u1));Vg̃(u1):=(g̃(u1)).
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
(2.4)Vh(u1,…,um)=(f̅(u1,…,um0,0,…,0m-m0︸),u1,…,um-1)+(f̃(u1,…,um̃)+g(u1,…,umg)+g̃(u1,…,um̃g),0,…,0m-1︸),
where f̅:(Df∪Df̃)⊂Rm→R is defined from f:Df⊂Rm0→R by adding (m-m0) identically zero arguments. The set D can be identical (although it is non-necessarily identical) to Df 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)
x̅0 is an equilibrium point of the uncontrolled nominal difference equation xn=fn(xn-1,…,xn-m0) if and only if x̅0=fn(x̅0,…,x̅0) for all n∈N. Then, X̅0=(x̅0,…,x̅0) is the associate equilibrium point of the first-order autonomous m0-order vector equationXn=Vfn(Xn-1); for all n∈N obtained from the particular difference equation (1.1) xn=fn(xn-1,…,xn-m0) for all n∈N via the nominal vector equation Vfn(u1,…,um)=(f(u1,…,um0),u1,…,um-1) provided that Vfn(Df)⊆Df. A sequence solution (x̅10,…,x̅m00) of xn=fn(xn-1,…,xn-m0) is a limit oscillatory solution of order at most m0 if and only if x̅km0+i0=fn(x̅10,…,x̅m00) for all k,n∈N for all i∈m̅0:={1,2,…,m0}. Such a solution is trivially an equilibrium point if x̅i0=x̅0 for all i∈m̅0. The m0-real vector X̅0=(x̅10,…,x̅m00) is the associate nominal limit oscillatory solution of order at most m0 of the first-order autonomous m0-order vector equation Xn=Vfn(Xn-1) for all n∈N obtained from the particular difference equation (1.1) xn=fn(xn-1,…,xn-m0) for all n∈N via the nominal vector equation Vfn(u1,…,um)=(f(u1,…,um0),u1,…,um-1);(2)
x̅0p is an equilibrium point of the uncontrolled perturbed difference equation xn=fn(xn-1,…,xn-m0)+f̃n(xn-1,…,xn-m̃) if and only if x̅0p=fn(x̅0p,…,x̅0p)+f̃n(x̅0p,…,x̅0p) for all n∈N. Then, X̅0p=(x̅0p,…,x̅0p) is the associate equilibrium point of the first-order autonomous m0p:=max(m0,m̃)-order vector equation Xn=Vfn+f̃n(Xn-1) for all n∈N according to:
(2.5)Vf+f̃(u1,…,um0p)=(f̅0(u1,…,um0p,0,…,0m0p-m0︸),u1,…,um0p-1)+(f̃̅0(u1,…,um̃,0,…,0m0p-m̃︸),0,…,0m0p-1︸,)
provided that Vfn+f̃n(Df∪Df̃)⊆Df∪Df̃  provided the set union is nonempty, where f̅0,f̃̅:(Df∪Df̃)⊂Rm0p→R in (2.5), provided that Df∪Df̃  is non-empty, take into account that the uncontrolled and nominal perturbed difference equations have potentially distinct orders and are built fromf:Df⊂Rm0→R, f̃:Df̃⊂Rm̃→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 m0p, or equivalently those of its associate vector function, by a sequence solution:
(2.6)x̅km0p+i0p=fn(x̅10p,…,x̅m0p0p)+f̃n(x̅10p,…,x̅m0p0p);∀k,n∈N,∀i∈m̅0p;(3)
x̅c is an equilibrium point of the controlled nominal difference equation xn=fn(xn-1,…,xn-m0)+gn(xn-1,…,xn-mg) if and only if x̅c=fn(x̅c,…,x̅c)+gn(x̅c,…,x̅c) for all n∈N. Then, X̅c=(x̅c,…,x̅c) is the associate equilibrium point of the first-order autonomous mc:=max(m0,mg)-order vector equation Xn=Vfn+gn(Xn-1) for all n∈N, provided that Vf+g(Df∪Dg)⊆Df∪Dg, provided that such a union is non-empty. The equivalent first-order vector equations are defined via the associate vector function Vf+g(Df∪Dg) through ad hoc functions f̅c:D→Df and g̅c:D→Dg 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 mc of the controlled nominal difference equation, which can be equivalently expressed in vector form, in the same way as above;(4)
x̅cp is an equilibrium point of the controlled perturbed difference equation (1.1) if and only if(2.7)x̅cp=hn(x̅cp,…,x̅cp)=fn(x̅cp,…,x̅cp)+f̃n(x̅cp,…,x̅cp)+gn(x̅cp,…,x̅cp)+g̃n(x̅cp,…,x̅cp);∀n∈N.
Then, X̅cp=(x̅cp,…,x̅cp) is the associate equilibrium point of the first-order autonomous m-order vector equation Xn=Vhn(Xn-1) for all n∈N defined via (2.4) provided that Vhn(D)⊆D. A sequence solution (x̅1cp,…,x̅mcp) of xn=fn(xn-1,…,xn-m) is a limit oscillatory solution of order at most m if and only if x̅km+icp=fn(x̅1cp,…,x̅mcp) for allk,n∈N for all i∈m̅. Such a solution is trivially an equilibrium point if x̅icp=x̅cp for all i∈m̅. The m-real vector X̅cp=(x̅1cp,…,x̅mcp) is the associate nominal limit oscillatory solution of order at most m of the first-order autonomous m-order vector equation Xn=Vfn(Xn-1) for all n∈N obtained from the particular difference equation (1.1) xn=fn(xn-1,…,xn-m) for all n∈N via the nominal vector equation Vfn(u1,…,um)=(f(u1,…,um),u1,…,um-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 equationxn=fn(xn-1,…,xn-m) for all n∈N has been pointed to be equivalent to its associate vector equation Xn=Vfn(Xn-1) for all n∈N. Then, the nominal uncontrolled difference equation admits the representation xn=fn(Xn-1)=fn(Vfn-1(Xn-2)) for all n∈N. Proceeding recursively:
(2.8)xn=fn(Xn-1)=fn(Vfn-1(Xn-2))=fn(Gn-1f(X0));∀n∈N.
By defining Gnf:=VfnoVfn-1o···oVf1 for all n∈N0  with G0f  being identity. Close composed mappings to describe the various uncontrolled and controlled (nominal or) versions of (1.1) are
(2.9)G0f+f̃=id,Gnf+f̃:=Vfn+f̃noVfn-1+f̃n-1o⋅⋅⋅oVf1+f̃1;∀n∈N,G0f+f̃+g=id,Gnf+f̃+g:=Vfn+f̃n+gnoVfn-1+f̃n-1+gn-1o⋅⋅⋅oVf1+f̃1+g1;∀n∈N,G0h=id,Gnh:=VhnoVhn-1o⋅⋅⋅oVh1;∀n∈N.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 ifm=m0=max(m0,m̃). 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,
(2.10)m0=max⁡(m0,mg,m̃g)=max⁡(m0,m̃g).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 mostm.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)
mp:=max(m0,m̃)=m0, Vfn(Df)⊆Df≠∅ and Vfn+f̃n(Df∪Df̃)⊆Df∪Df̃≠∅ for all n∈N, where Vfn+f̃n(u1,…,ump) is defined in (2.5) for all n∈N;(2)
∂fn(X)/∂XT|X̅0, ∂f̃n(X)/∂XT|X̅0 exist within a neighborhood of X̅0, which is a limit oscillatory solution of the vector uncontrolled nominal equation of order at most mp=m0 (including potentially nominal equilibrium points as particular cases);(3)
the inversem0-matrix (Imp-M0(X̅0))-1 exists, where M0(X̅0)=∂Vf(X)/∂XT|X̅0+∂Vf̃(X)/∂XT|X̅0, and Im0 is the m0-identity matrix of Rm0.Then, X̅̂0p=X̅0+(Im-M0(X̅0))-1Vf̃(X̅0) 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 X̅0p as ∥Vf̃(X̅0)∥ is sufficiently small.
Ifrank(Imp-M0(X̅0),Vf̃(X̅0))=rank(Imp-M0(X̅0))<mp, then there are infinitely many first-order estimates X̅̂0p of limiting oscillatory solutions of the vector uncontrolled nominal equation of order at most m0. If mp-1≥rank(Im-M0(X̅0)andVf̃(X̅0))>rank(Imp-M0(X̅0)), then there is no such an estimate.Proof.
Note thatmp=m0 implies that m̃≤m0. Define ΔX̅0p,0:=X̅0p-X̅0 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:
(2.11)ΔX̅n+10p,0=Mn0(X̅0)ΔX̅n0p,0+Vf̃n(X̅0)+o(‖ΔX̅n0p,0‖)Imp=M0(X̅0)ΔX̅n0p,0+Vf̃(X̅0)+o(‖ΔX̅n0p,0‖)Imp;∀n∈N,
where
(2.12)Mn0(X̅0)=∂Vfn(X)∂XT|X̅0+∂Vf̃n(X)∂XT|X̅0=[∂fn(X)∂XT|X̅0+∂f̃n(X)∂XT|X̅0Imp-10]=M0(X̅0)=∂Vf(X)∂XT|X̅0+∂Vf̃(X)∂XT|X̅0=[∂f(X)∂XT|X̅0+∂f̃(X)∂XT|X̅0Im-10];∀n∈N,
if m0≥2, where Im0-1 is the (m0-1) identity matrix and superscript T denotes transposition, since Vfn(X̅0)=Vf(X̅0), Vf̃n(X̅0)=Vf̃(X̅0), fn(X̅0)=f(X̅0), f̃n(X̅0)=f̃(X̅0) for all n∈N;
(2.13)Mn0(X̅0)=∂Vfn(X)∂X|X̅0+∂Vf̃n(X)∂X|X̅0=M0(X̅0)=∂Vf(X)∂X|X̅0+∂Vf̃(X)∂X|X̅0;∀n∈N,
if m0=1. Taking ΔX̅n+10p,0=ΔX̅n0p,0=ΔX̅0p,0 for all n∈N, one gets from (2.11):
(2.14)ΔX̅0p,0=(Im0-M0(X̅0))-1Vf̃(X̅0)+o(‖ΔX̅0p,0‖)Im0,
provided that (Im0-M0(X̅0))-1 exists so that ΔX̅̂0p,0=(Im-M0(X̅0))-1Vf̃(X̅0) is an estimate of ΔX̅n+10p,0 so that if ∥M0(X̅0)∥<1 and ∥Vf̃(X̅0)∥<(1-∥M0(X̅0)∥)ε for some ε∈R+, then one gets from Banach’s perturbation lemma [24]:
(2.15)‖ΔX̅̂0p,0‖≤‖(Im-M0(X̅0))-1‖‖Vf̃(X̅0)‖≤‖Vf̃(X̅0)‖1-‖M0(X̅0)‖<ε.
Since for ε=0, Vf̃(X̅0)=ΔX̅0p,0=0 then, for a sufficiently small ε* such that ∥Vf̃(X̅0)∥/(1-∥M0(X̅0)∥)<ε* and for any ε≤ε*, o(∥ΔX̅0p,0∥)≤ε/2 what occurs in particular, for ε<1 if f,f̃:Rm→R are furthermore analytic in an open ball of Rm centred at X̅0 of radius ρ=3ε/2. The conditions for the existence of infinitely many first-order estimates of ΔX̅̂0p,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 Theorem2.5 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,m̃,mg) and m̃g≤max(m0,m̃,mg), so that m=max(m0,m̃,mg,m̃g), and Vhn(D)⊆D≠∅, Vfn+f̃n+gn(Df∪Df̃∪Dg)⊆Df∪Df̃∪Dg≠∅ for alln∈N; where Vfn+f̃n+gn(u1,…,um) is defined correspondingly to (2.5) for this case for all n∈N;(2)
∂(fn(X)+f̃n(X)+gn(X))/∂XT|X̅c, ∂g̃n(X)/∂XT|X̅c exist within a neighborhod of X̅c, 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(Im-Mc(X̅c))-1 exists, where Mc(X̅c)=∂Vh(X)/∂XT|X̅c.Then X̅̂cp=X̅c+(Im-Mc(X̅c))-1Vg̃(X̅c) 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 ∥Vg̃(X̅c)∥ is sufficiently small.
Ifrank(Im-Mc(X̅c),Vg̃(X̅c))=rank(Im-Mc(X̅c))<m, then there are infinitely many first-order estimates X̅̂cp of limiting oscillatory solutions of the vector uncontrolled nominal equation of order at most m. If m-1≥rank(Im-Mc(X̅c),Vf̃(X̅c))>rank(Im-Mc(X̅c)), 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)
mc:=max(m0,mg)=m0, Vfn(Df)⊆Df≠∅ and Vfn+gn(Df∪Dg)⊆Df∪Dg≠∅; foralln∈N, where Vfn+gn(u1,…,umc) is defined correspondingly to (2.5) for this case for all n∈N;(2)
∂fn(X)/∂XT|X̅0, ∂gn(X)/∂XT|X̅0 exist within a neighborhod of X̅0, 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 inversem0-matrix (Im0-Mc(X̅0))-1 exists, where Mc(X̅0)=∂Vf(X)/∂XT|X̅0+∂Vg(X)/∂XT|X̅0, and Im0  is the m0-identity matrix of Rm0.Then, X̅̂c=X̅0+(Im-Mc(X̅0))-1Vg(X̅0) 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 ∥Vg(X̅0)∥ is sufficiently small.
Ifrank(Im0-Mc(X̅0),Vg(X̅0))=rank(Im0-Mc(X̅0))<m0, then there are infinitely many first-order estimates X̅̂c of limiting oscillatory solutions of the vector controlled nominal equation of order at most mc=m0. If m0-1≥rank(Im0-Mc(X̅0),Vg(X̅0))>rank(Im0-Mc(X̅0)), then there is no such an estimate.Theorem 2.8.
Assume the following:(1)
mcp:=max(m0,m̃,mg)=max(m0,m̃)=m0, Vfn(Df∪Df̃)⊆Df∪Df̃≠∅, and Vfn+f̃n+gn(Df∪Df̃∪Dg)⊆Df∪Df̃∪Dg≠∅ for all n∈N, where Vfn+f̃n+gn(u1,…,umcp) is defined correspondingly to (2.5) for this case for all n∈N;(2)
∂fn(X)/∂XT|X0, ∂f̃n(X)/∂XT|X0, ∂gn(X)/∂XT|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 inversem0-matrix (Im0-Mcp(X0))-1 exists, where Mcp(X0)=∂Vf(X)/∂XT|X0+∂Vf̃(X)/∂XT|X0+∂Vg(X)/∂XT|X0, and Im0 is the m0-identity matrix of Rm0.
Then,X̅̂cp=X0+(Im-Mcp(X0))-1Vf̃+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 ∥Vf̃+g(X0)∥ is sufficiently small.
Ifrank(Im0-Mcp(X0),Vf̃+g(X0))=rank(Im0-Mcp(X0))<m0 then there are infinitely many first-order estimates X̅̂cp of limiting oscillatory solutions of the vector controlled nominal equation of order at most mcp=m0. If m0-1≥rank(Im0-Mcp(X0),Vf̃+g(X0))>rank(Im0-Mcp(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)
letx̅0and x̅c be two equilibrium points of the nominal and nominal controlled difference equations with corresponding ones X̅0 and X̅c in the associate vector equations. Assume that m=m0=mc, X̅0∈clS∩A0 and X̅c∈clS∩Ac with ∅≠S0⊆S⊆A0∩Ac, where Sis an invariant subset of solutions of the associate vector equations for all Vfn and Vfn+gn, that is, Vfn(S)⊆Sand Vfn+gn(S)⊆S, for any controller in C consisting in a nominal controller, with
(3.1)A0:={X∈Rm:|fn(X)-x̅0|≥αn‖X-X̅0‖;∀n∈N},Ac:={X∈Rm:|fn(X)+gn(X)-x̅c|≤βnc‖X-X̅c‖;∀n∈N},
for some real nonnegative sequences {αi}i∈Nand {βic}i∈N. If {α̅n}n∈N is unbounded, where α̅n=∏i=1nαi, and {β̅nc}n∈N is such that limsupn→∞β̅nc<1, where β̅nc=∏i=1nβic, then X̅0 is unstable where X̅c is locally asymptotically stable with respect to S;(ii)
letx̅0p and x̅c be two equilibrium points of the uncontrolled perturbed and nominal controlled (via the nominal plus the incremental controllers) difference equations with corresponding ones X̅0p and X̅cin the associate vector equations. If m=m0p=max(m0,m̃)=mcp, X̅0p∈clS∩A0p and X̅c∈clS∩Ac with ∅≠S0⊆S⊆A0p∩Ac, where S is an invariant subset of the solutions of the associate vector equations for all Vfn+f̃n and Vfn+f̃n+gn; that is, Vfn+f̃n(S)⊆S and Vfn+f̃n+gn(S)⊆S, with
(3.2)A0p:={X∈Rm:|fn(X)+f̃n(X)-x0p̅|≥αpn‖X-X̅0p‖;∀n∈N},Ac:={X∈Rm:|fn(X)+f̃n(X)+gn(X)-x̅c|≤βnc‖X-X̅c‖;∀n∈N},
for some nonnegative sequences {αpi}i∈N and {βic}i∈N. Thus, if {α̅pn}n∈N is unbounded, where α̅pn=∏i=1nαpi, and {β̅nc}n∈N is such that limsupn→∞β̅nc<1, where β̅nc=∏i=1nβic, then X̅0p is unstable where X̅c is locally asymptotically stable with respect to S;(iii)
letx̅0p and x̅cp be two equilibrium points of the perturbed and perturbed controlled (via the nominal plus the incremental controllers) difference equations with corresponding ones X̅0p and X̅cp in the associate vector equations. If m=m0p=mcp, X̅0p∈clS∩A0p, and X̅cp∈clS∩Acp with ∅≠S0⊆S⊆A0p∩Acp, where S is an invariant subset of the solutions of the associate vector equations for all Vfn+f̃n and Vhn, that is, Vfn+f̃n(S)⊆S and Vhn(S)⊆S, for some nonnegative real sequences {αi}i∈N and {βpic}i∈N, and
(3.3)Acp:={X∈Rm:|hn(X)-x̅cp|≤βpnc‖X-X̅cp‖;∀n∈N}.
Thus, if {α̅n}n∈N is unbounded, where α̅n=∏i=1nαi, and {β̅pnc}n∈N is such that limsupn→∞β̅pnc<1, where β̅pnc=∏i=1nβpic, then X̅0p is unstable where X̅cp 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 Section2. 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 thatm=m0p=mcp with X̅0p∈clS∩A0 and X̅cp∈clS∩Acp being unique equilibrium points in clS∩A0, respectively in clS∩Acp, where S⊆A0p∩Acp is invariant under all Vfn+f̃n and Vhn for a class of controllers C, that is, Vfn+f̃n(S)⊆S and Vhn(S)⊆S for any combined nominal plus incremental controller in the class C for all n∈N. Define Δx̅0p:=x̅cp-x̅0p and ΔX̅0p:=X̅cp-X̅0p being sufficiently close to zero to satisfy
(3.4)‖ΔX̅0p‖≤(βpnc)-1(αpn-βpnc)‖X-X̅0p‖,∀X∈cl⁡S,∀n∈N.
Assume also that the nominal and incremental controllers are
(3.5)gn(Xn-1)=λn(Xn-1)fn-σn(Xn-1-σn);g̃n(Xn-1)=λ̃n(Xn-1)f̃n-σ̃n(Xn-1-σ̃n);∀n∈N,
with their corresponding gain sequences {λn}n∈N and {λ̃n}n∈N being chosen to satisfy the constraints:
(3.6)sign⁡λn(Xn-1)=-sign⁡((fn(Xn-1)-x̅0p)fn-σn(Xn-1-σn));∀n∈N,(3.7)sign⁡λ̃n(Xn-1)=-sign⁡((f̃n(Xn-1)-x̅0p)f̃n-σ̃n(Xn-1-σ̃n));∀n∈N,(3.8)‖λn(Xn-1),λ̃n(Xn-1)‖≤|fn(Xn-1)+f̃n(Xn-1)-x̅cp|‖(fn-σn(Xn-1-σn),f̃n-σ̃n(Xn-1-σ̃n))‖;∀n∈N,
for some existing nonnegative integer sequences {σn}n∈N0, {σ̃n}n∈N0 chosen such that ∥(fn-σn(Xn-1-σn),f̃n-σ̃n(Xn-1-σ̃n))∥≠0 for all n∈N0 subject to liminfn→∞∥(fn-σn(Xn-1-σn),f̃n-σ̃n(Xn-1-σ̃n))∥≥0since equality to zero holds for all nonnegative sequences {σn}n∈N0, {σ̃n}n∈N0 if and only if X̅cp=0.
Then, the corresponding equilibrium point of the perturbed uncontrolled associate vector systemX̅0p is unstable, while that of the perturbed controlled system X̅cp is asymptotically stable. The properties hold for the corresponding perturbed and perturbed controlled difference equations of equilibrium points x̅0p and x̅cp, respectively.Proof.
Conditions for the following chain of inequalities to hold are given:(3.9)|hn(Xn-1)-x̅cp|=|fn(Xn-1)+f̃n(Xn-1)+gn(Xn-1)+g̃n(Xn-1)-x̅0p-Δx̅0p|≤βpnc‖Xn-1-X̅c‖≤αpn‖Xn-1-X̅0p‖≤|fn(Xn-1)+f̃n(Xn-1)-x̅0p|;∀n∈N
within S. The following chained inequalities guarantee that (3.9) holds in S:
(3.10)αpn‖X-X̅0p‖-|λn(Xn-1)fn-σn(Xn-1-σn)+λ̃n(Xn-1)f̃n-σ̃n(Xn-1-σ̃n)-Δx̅0p|≤|fn(Xn-1)+f̃n(Xn-1)-x̅0p+λn(Xn-1)fn-σn(Xn-1-σn)+λ̃n(Xn-1)f̃n-σ̃n(Xn-1-σ̃n)-Δx̅0p|=|fn(Xn-1)+f̃n(Xn-1)-x̅0p-Δx̅0p|-|λn(Xn-1)fn-σn(Xn-1-σn)+λ̃n(Xn-1)f̃n-σ̃n(Xn-1-σ̃n)|≤βpnc‖X-X̅0p-ΔX̅0p‖≤αpn‖X-X̅0p‖;∀n∈N,0≤|fn(Xn-1)+f̃n(Xn-1)-x̅0p-Δx̅0p|-βpnc‖X-X̅0p-ΔX̅0p‖≤|λn(Xn-1)fn-σn(Xn-1-σn)+λ̃n(Xn-1)f̃n-σ̃n(Xn-1-σ̃n)|;∀n∈N.
Then, the nominal and incremental controller gains are chosen to satisfy (3.6)–(3.8) for existing nonnegative real sequences {σn}n∈N0, {σ̃n}n∈N0 such that ∥(fn-σn(Xn-1-σn),f̃n-σ̃n(Xn-1-σ̃n))∥≠0 for all n∈N0 subject to liminfn→∞∥(fn-σn(Xn-1-σn),f̃n-σ̃n(Xn-1-σ̃n))∥≥0since equality to zero holds if and only if X̅cp=0. The reminder of the proof now follows since from (3.9) one gets simultaneously within a nonempty invariant set S⊆A0p∩Acp:
(3.11)|hn(Xn-1)-x̅cp|≤βpnc‖Xn-1-X̅cp‖;∀Xi∈S,∀i∈N0,(3.12)αpn‖Xn-1-X̅0p‖≤|fn(Xn-1)+f̃n(Xn-1)-x̅0p|;∀Xi∈S,∀i∈N0.
Then, one gets from (3.12):
(3.13)|hn(X)-x̅cp|=|hn(Vhn-1(X))-X̅cp|=|hn(Gn-1h(X))-x̅cp|≤βpnc‖X-X̅cp‖≤∏i=1n[max⁡i≥j(∈N)≥i-m-1(βpjc)]×‖X-X̅cp‖≤β̅cn‖X-X̅cp‖;∀X∈cl⁡S,∀n∈N,
with the real sequence {β̅cn}n∈N0 of elements satisfying β̅cn∈[0,1), foralln∈N so that one deduces by taking ℓ∞-norms for the m-tuples G(·)h(X) (see Remark 2.3) that:
(3.14)‖Gn-1h(X)-X̅cp‖∞=max⁡n≥j(∈N)≥n-m-1|hj(Gj-1h(X))-x̅cp|≤β̅cn‖X-X̅cp‖∞;∀n∈N.
Since β̅cn∈(0,1) for all n∈N, then limn→∞(Gn-1h(X)-X̅cp)=0 for all X∈S. Since {Gnh(X)}n∈N is a sequence of contraction self-mappings from Rm∣S to S, and Rm is a complete metric space endowed with the given norm-induced metric, then the equilibrium point X̅cp 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 X̅cp on the controlled difference equation is locally asymptotically stable with respect to S, and the equilibrium point x̅cp of the corresponding difference equation is also locally stable. On the other hand, it follows from (3.12) that
(3.15)αpn-1‖X-X̅0p‖∞=αpn‖Gn-1f+f̃(X)-X̅0p‖∞≤|fn(X)+f̃n(X)-x̅0p|=|fn(Vfn+f̃n(X))+f̃n(Vfn+f̃n(X))-x̅0p|≤‖Gnf+f̃(X)-X̅0p‖∞;∀X∈cl⁡S,∀n∈N.
Since {α̅pn}n∈N is unbounded with α̅pn:=∏i=1nαpi, then {Gnf+f̃(X)}n∈N is a sequence of expanding self-mappings from Rm∣S to S so that the equilibrium point X̅0p 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 x̅0p 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=m0p=mc with X̅0p∈clS∩A0p, and X̅c∈clS∩Ac being unique equilibrium points in clS∩A0p, respectively in clS∩Ac, where S⊆A0p∩Ac is invariant under all Vfn+f̃n, and Vhn≡Vfn+f̃n+gn (since g̃n≡0) for any controller in the class C consisting of a nominal controller, that is, Vfn+f̃n(S)⊆S and Vhn(S)⊆S with the incremental controller being identically zero for all n∈N, provided that the sets are nonempty. Assume also that the sets A0p and Ac defined in (3.2) are redefined as
(3.16)A0p:={X∈Rm:|fn(X)+f̃n(X)-x̅0p|≥αpn(1-α̃pn)‖X-X̅0p‖;∀n∈N},Ac:={X∈Rm:|fn(X)+f̃n(X)+gn(X)-x̅c|≤βnc(1+β̃nc)‖X-X̅c‖;∀n∈N},
for nonnegative real sequences {αpn(1-α̃pn)}n∈N and {βnc(1+β̃nc)}n∈N being defined for some nonnegative real sequences {α̃pn}n∈N and {β̃nc}n∈N, subject to α̃pn≤1 and β̃nc<βn-1-1, where {α̅pn}n∈N is unbounded of elements redefined as α̅pn:=∏i=1n[αpi(1-α̃pi)], and {β̅nc}n∈N of elements being redefined as β̅nc:=∏i=1n[βic(1+β̃ic)] being such that limsupn→∞β̅nc<1;(2)
the perturbed sequence{f̃n(Xn-1)}n∈N satisfies the constraints:
(3.17)|f̃n(Xn-1)-an|≤βncβ̃nc‖Xn-1-X̅c‖;|f̃n(Xn-1)-bn|≤αpnα̃pn‖Xn-1-X̅0p‖,
for all n∈N within S for some real sequences {an}n∈Nand {bn}n∈N and for some nonnegative real sequences {α̃pn}n∈N0 and {β̃nc}n∈N0, subject to α̃pn≤1 and β̃nc<βn-1-1;(3)
the stabilizing incremental controller is identically zero, while the nominal controller isgn(Xn-1)=λn(Xn-1)fn-σn(Xn-1-σn) subject to
(3.18)sign⁡λn(Xn-1)=-sign⁡((fn(Xn-1)-x̅0p)fn-σn(Xn-1-σn));∀n∈N,|λn(Xn-1)|≤|fn(Xn-1)-x̅c+an||fn-σn(Xn-1-σn)|;∀n∈N,
within S if fn-σn(Xn-1-σn)≠0 and λn(Xn-1)=0 if fn-σn(Xn-1-σn)=0 for all n∈N, for some existing non-negative integer sequence {σn}n∈N0.
Then, the corresponding equilibrium point of the perturbed uncontrolled associate vector systemX̅0p is unstable, while that of the controlled system X̅c under the nominal controller is asymptotically stable. The stability properties also hold for the corresponding perturbed and perturbed controlled difference equations of equilibrium points x̅0p and x̅c, respectively.Proof.
Now the errors of the uncontrolled perturbed and the controlled nominal equilibrium points under consideration areΔx̅0p:=x̅c-x̅0p and ΔX̅0p:=X̅c-X̅0p. Note that g̃n(Xn-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
(3.19)|fn(Xn-1)+f̃n(Xn-1)+λn(Xn-1)fn-σn(Xn-1-σn)-x̅c|≤|fn(Xn-1)+λn(Xn-1)fn-σn(Xn-1-σn)-x̅c+an|+|f̃n(Xn-1)-an|≤βnc(1+β̃nc)‖Xn-1-X̅c‖;∀Xn∈S,∀n∈N0,(3.20)αpn(1-α̃pn)‖Xn-1-X̅0p‖≤|fn(Xn-1)-x̅0p+bn|-|f̃n(Xn-1)-bn|≤|fn(Xn-1)+f̃n(Xn-1)-x̅0p|;∀Xn∈S,∀n∈N0,
for all n∈N0, provided that
(3.21)|fn(Xn-1)+λn(Xn-1)fn-σn(Xn-1-σn)-x̅c+an|≤βnc‖Xn-1-X̅c‖;∀Xn∈S,∀n∈N0,αpn‖Xn-1-X̅0p‖≤|fn(Xn-1)-x̅0p+bn|;∀Xn∈S,∀n∈N0,
for some existing nonnegative sequences {αpn(1-α̃pn)}n∈N and {βnc(1+β̃nc)}n∈N. Thus, X̅0p is unstable, while X̅cis locally asymptotically stable with respect to S. Those properties also hold by construction of X̅0p and X̅c for the corresponding perturbed and perturbed controlled difference equations of equilibrium points x̅0p and x̅c, 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 Section2 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:
(3.22)Δ:=X̅̂c-X0+(Im-Mc(X0))-1Vf̃+g(X0),
for all X∈S calculated from Theorem 2.8. Thus, the last term of (3.19) with β̃nc=0 for all n∈N0 possesses the lower-bounds given below:
(3.23)βnc‖X-X̅c‖=βnc‖X-X̅̂c-Δ‖≥βnc‖X-X̅̂c‖-βnc‖Δ‖≥βnc(1-εnc)‖X-X̅̂c‖,
provided that the equilibrium point estimation error is of sufficiently small size fulfilling ∥Δ∥≤εnc∥X-X̅̂c∥ for some sequence {εnc}n∈N0 satisfying 0≤εnc≤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:(4.1)xn+1=hn(xn)=fn(xn)+f̃n(xn)+gn(xn)=(δn+δ̃n)xne-xn+gn(xn),
where δn≥1 for all n∈N, fn(xn)=δnxne-xn is the uncontrolled nominal dynamics, f̃n(xn)=δ̃nxne-xn is the unmodeled uncontrolled dynamics, and gn(xn) 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 x̅=0. If δn→δ>1 as n→∞, then x̅=lnδis also an equilibrium point of such an equation.The particularxn+1=hn(xn)=fn(xn) is the nominal uncontrolled part of (4.1). If δn≥1 for all n∈N, S≡R—(the set of negative real numbers) and A0:={x∈R:|fn(x)|≥δn|x|}, then S⊂A0 is invariant.The equilibrium pointx̅=0  of the uncontrolled nominal difference equations is unstable with respect to S;  x0=0⇒xn=0; for alln∈N, and if x0>0 and finite, then{xn}n∈N is nonnegative and uniformly bounded. To see this property, proceed with a contradiction argument as follows. Assume that {xn}n∈N is unbounded, then if {xn}n∈N is unbounded, then there exists a subsequence {xn}n∈N̂ of it which diverges, that is, xn→+∞ as n(∈N̂)→∞ with N̂⊂N is numerable of infinite cardinal. As a result, there is a monotone increasing real sequence {Mn}n∈N̂ such that Mn+1≥xn>Mn so that Mn+1≤xn+1≤δnxne-Mn≤δnMn+1e-Mn for all n∈N̂ so that 1≤liminfn(∈N̂)→∞δne-Mn=0, which is clearly a contradiction to {xn}n∈N being unbounded by assuming that {δn}n∈N̂ is not exponentially unbounded. Then, {xn}n∈N is bounded if x0≥0 and δn≥1 for all n∈N. The above contradiction argument does not require specifically δn≥1, but only δn>0 for all n∈N. Thus, {xn}n∈N is bounded if x0≥0 and δn≥0 for all n∈N, and then the equilibrium point x̅=0 of the uncontrolled nominal difference equations is locally stable with respect to the complement S̅ 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 δn≥1 for all n∈N and globally stable with respect to S̅ if δn>0 for all n∈N. A combined result is x̅=0 is a semistable equilibrium point if δn≥1.One now considers the synthesis of a stabilizing controllergn(xn)=λnδnxne-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 xn+1=hn(xn) is still x̅=0 as for the uncontrolled nominal and perturbed ones. Equation (4.1) becomes(4.2)xn+1=hn(xn)=((1+λn)δn+δ̃n)xne-xn;n∈N0.
It is required from Theorem 3.3, (3.18) with δn=αpn, σn=0, that  λn<0 if x0>0 and λn>0 if xn<0 and for some real sequence {an}n∈N0, and a gain sequence satisfying:(4.3)|λn|=|δnxn+anexnδnxn|={|1+anexnδnxn|ifx0≥0withan<0,|an|≥|δnxne-xn|;|1-anδn|xn|e|xn||ifxn<0with|an|≥δn|xn|e|xn|;∀n∈N0.
Thus,(a)
ifx0≥0, then |1+anexn/δnxn|=|anexn/δnxn|-1 so that one gets from (4.2):
(4.4)xn+1=|an|+δ̃nxne-xn<xne-xn<xnsincexn>0;∀n∈N0,
provided that the controller gain sequence and unmodeled dynamics contribution sequence satisfy:
(4.5)λn=-(1-|an|exnδnxn),|an|≥|δnxne-xn|;∀n∈N0,(4.6)δ̃n∈[δ̂1n,δ̂2n)≡[-|anexnxn|,1-anexnxn)⊆[-δn,1-δn);∀n∈N0,
where f̃n(xn)=δ̃nxne-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)
ifx0<0 then for any n∈N0 such that xn<0, and |1+anexn/δnxn|=|anexn/δnxn|-1 so that one gets from (4.2) that
(4.7)xn+1=-(an+δ̃n|xn|e|xn|)<-|xn|≤-βn|xn|;∀n∈N0,
for any sequence {βn}n∈N0 with any βn<1 for all n∈N0 provided that the controller gain sequence and unmodeled dynamics contribution sequence satisfy(4.8)λn=|1-anδn|xn|e|xn||,|xn|≥an>0;∀n∈N0,(4.9)δ̃n≤(an|xn|-1)e|-xn|;∀n∈N0.
Note that for both cases (a)-(b), |(1+λn)δn+δ̃n|<1 for all n∈N0, and {xn}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 {δn}n∈N0. Theorem 3.3 is fulfilled also for the sets (3.16) with α̃pn=β̃nc=0, αpn=δn≥1 and βnc<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 pointx̅=0 of the difference equation of Section 4 is globally asymptotically stable around the equilibrium point x̅=0 (so that it is asymptotically stable related to both the invariant set S and its complement in R) under the feedback controller gn(xn)=λnδnxne-xn for all n∈N0 given by the gain sequence of elements satisfying (4.5) if x0≥0 (implying that xn≥0 for all n∈N) and (4.8) if xn<0 for the current n∈N0 for any structured unmodeled dynamics f̃n(xn)=δ̃nxne-xn, subject to (4.6) if x0≥0, and to (4.9) if x0<0 provided that xn<0 for the current n∈N0.



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*Source: 320302-2012-04-01.xml*