# Controllability and Observability Analysis of Nonlinear Positive Discrete Systems

Authors: Mouhcine Naim; Fouad Lahmidi; Abdelwahed Namir
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This article studies the controllability and observability of nonlinear positive discrete systems. These properties play a fundamental role in system analysis before controller and observer design is engaged. We solve these problems by a technique based on the fixed point theory.


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## Body

## 1. Introduction

Controllability and observability are two fundamental concepts in the mathematical control theory. Systematic studies on these topics in the linear case were started at the beginning of 1960s [1, 2] and in nonlinear one in 1970s [3]. Controllability property plays an important role in the existence of solutions to many control problems, for example, stabilization of unstable system by feedback, optimal control [4]. Observability plays a crucial role in the study of canonical forms of dynamical systems or observer synthesis [5]. Basically a system is controllable if it is possible to transfer it from an arbitrary initial state to an arbitrary final state using only certain admissible controls; it is observable if the initial state can be determined using the information given by an output over a finite time. There exist many papers in which these two properties for classical discrete and continues systems are studied. A meaningful fact in practice, also in classical systems, is to investigate both properties in their local formulations for nonlinear systems through global notions of controllability and observability by linearisation of the considered systems.Positive systems are a wide class of systems in which state variables and outputs are constrained to be positive, or at least nonnegative, for all time whenever the initial state and inputs are nonnegative. Since the state variables and outputs of many real-world processes represent quantities that may not have meaning unless they are nonnegative because they measure concentrations, temperatures, cell birth, losses, etc., positive systems arise frequently in mathematical modeling of engineering problems, management sciences, economics, social sciences, chemistry, biology, ecology, medicine, and other areas.The mathematical theory of positive linear systems is based on the theory of nonnegative matrices developed by Perron and Frobenius; see, for example, [6, 7]. An excellent survey of positive systems with an emphasis on their applications in the areas of management and social sciences is given by Luenberger in [6]. The more recent monographs by Farina and Rinaldi in [8] and Kaczorek in [9] are devoted entirely to positive linear systems and some of their applications. Since positive systems are not defined on linear spaces but on cones [10, 11], their analysis and synthesis are more complicated and more challenging.Since late 1980s controllability and reachability of both discrete and continuous positive linear systems have been a subject of much research [12–19]. Therefore, it was discovered that controllability of continuous positive systems requires very restrictive conditions to be satisfied. Thus criteria for controllability of discrete systems and continuous systems are essentially different. Observability of positive linear systems has been addressed in [20, 21]. The reachability of nonlinear positive systems for continuous and discrete systems has been formulated and solved, respectively, in [22, 23].This paper deals with the class of nonlinear positive discrete systems. The problems of controllability and observability for this class of systems are considered. First, we present a characterization of the positivity of such systems, and then necessary and sufficient conditions for checking the controllability and observability properties are established. Because of the nonlinearity of the proposed system, we characterize these properties by two different methods that are mainly based on fixed point technique. We show that controllability is equivalent to existence of at least one fixed point and observability is equivalent to existence of at most one fixed point of some functions. Furthermore, we characterize the set of nonnegative controls which steer the state of the positive system from a nonnegative initial state to a nonnegative desired final state. The set of all nonnegative states which correspond to a given nonnegative output is also characterized. To the best of the author’s knowledge, the controllability and observability problems of nonlinear positive discrete systems have not been investigated yet by means of similar techniques as those presented in this work.


## 2. Preliminaries

First we introduce some notations.N is the set of nonnegative integers, N+ the set of positive integers, σsk=s,s+1,…,k the finite subset of N with s≤k, Rn the set of real vectors with n components, R+n the set of all vectors in Rn with nonnegative components, i.e.,(1)R+n≔x=x1,x2,…,xnT∈Rn:xi≥0,i∈σ1n,where T denotes the transpose, (zi)j the jth component of a vector zi, Rn×m the set of real constant matrices of size n×m, and In the identity matrix in Rn×n. In addition, if F is a mapping, then SF denotes the set of all its fixed points.In this work, we consider the discrete nonlinear system described by(2)xi+1=Axi+fxi+Bui,i∈N,x0∈Rn.Information on system (2) is given by the output equation(3)yi=Cxi,i∈N,where xi∈Rn is the system state, ui∈Rm is the input (or control), yi∈Rr, A∈Rn×n, B∈Rn×m, C∈Rr×n, f:Rn→Rn is a nonlinear function, and x0 represents the initial state.Now, we define the positivity of system (2)-(3) using the following.Definition 1.
System (2)-(3) is said to be positive if for any nonnegative initial state x0∈R+n and all nonnegative inputs ui∈R+m, i∈N, we have xi∈R+n and yi∈R+r for any i∈N, where xi is the solution corresponding to x0 and ui, and, similarly, yi is the output corresponding to x0 and ui.Definition 2.
A matrixA=aiji∈σ1n,j∈σ1m is said to be nonnegative and denoted by A∈R+n×m, if all of its elements are nonnegative, i.e., aij≥0 for all i∈σ1n,j∈σ1m.The following result provides a necessary and sufficient condition for positivity of system (2)-(3).Proposition 3.
System (2)-(3) is positive if and only if(4)A+fR+n⊂R+n,B∈R+n×m,C∈R+r×n.Proof.
See AppendixA.In the remainder of this paper, we assume that system (2)-(3) is positive.


## 3. Controllability of Nonlinear Positive Discrete Systems

The precise definition of the controllability of system (2) is given as follows.Definition 4.
System (2) is said to be controllable in N∈N+ steps if for any initial state x0∈R+n and any desired final state xf∈R+n, there exists an input sequence ui∈R+m,i∈σ0N-1, which steers the state of the system from x0 to xf, i.e., xN=xf.The explicit solution of (2) is given by(5)xi=Aix0+∑j=0i-1Ai-1-jfxj+∑j=0i-1Ai-1-jBuj,i∈N+.In this section, we will use the following notation:x=x1x2⋮xN∈RnN,u=u0u1⋮uN-1∈RNm, with N∈N+, xi∈Rn for i∈σ1N and ui∈Rm for i∈σ0N-1.We define the matrix(6)A~=AA2⋮AN∈RnN×n.Let us consider the nonlinear mappingG defined by(7)G:x∈R+nN↦Gx∈RnN,with(8)Gxi=∑j=0i-1Ai-1-jfxj,i∈σ1N,and let D denote the linear mapping defined by(9)D:u∈RNm↦Du∈RnN,with(10)Dui=∑j=0i-1Ai-1-jBuj,i∈σ1N.Then, solution (5) over N steps can be rewritten as(11)x=A~x0+Gx+Du.


### 3.1. Characterization of Controllability: First Mapping

The aim of this subsection is to establish a necessary and sufficient condition for the controllability of system (2) based on fixed points of a function appropriately chosen. Also, we characterize the set U+ of nonnegative controls which steer the state of system (2) from an initial state x0∈R+n at i=0 to a desired final state xf∈R+n at i=N∈N+, i.e.,(12)U+=u∈R+Nm:xNu=xf,where xNu is the state of system (2) in step N corresponding to the control u.Definition 5.
The positive image of a matrixE∈Rp×q is(13)Im+E=Eu∈Rp:u∈R+q.LetP:RnN→Im+D be any projection on Im+D (i.e., any mapping P:RnN→Im+D that satisfies Pz=z if and only if z∈Im+D) and x- be any fixed element of Im+D different from zero.We define(14)ζ:x∈R+nN↦x-A~x0-Gx∈RnN,and we consider the mapping(15)F:x∈R+nN↦A~x0+Gx+Pζx+xN-xfx-∈RnN,where · is a norm on Rn. It is to be noted that the mapping F depends on the states x0 and xf.A necessary and sufficient condition for the controllability of system (2) is given by the following.Proposition 6.
The nonlinear system (2) is controllable in N steps if and only if for all x0,xf∈R+n,F has a fixed point.Proof.
See AppendixB.Remark 7.
The fixed points ofF are independent of the choice of the projection operator P and the element x-. Indeed, let P1 and P2 be two projections on Im+D and x-1 and x-2 be two any elements not equal to zero of Im+D. Let us consider the mappings(16)F1:x∈R+nN↦A~x0+Gx+P1ζx+xN-xfx-1∈RnN,and(17)F2:x∈R+nN↦A~x0+Gx+P2ζx+xN-xfx-2∈RnN.
Letx be a fixed point of F1. By proof of Proposition 6, we have xN=xf and ζx∈Im+D. Consequently P2ζx=ζx and xN-xf=0, and then(18)F2x=A~x0+Gx+ζx=x.
Hence, ifx is a fixed point of F1, then it is also a fixed point of F2.Remark 8.
Proposition6 is still true if the expression xN-xf is substituted by the term gxN, where g:Rn→R+ is any function which satisfies gz=0 if and only if z=xf.Now, a characterization of the setU+ is given by the following result.Proposition 9.
We have(19)U+=D†ζx+Ψ∈R+Nm:x∈SFandΨ∈ker⁡D,with D† being the pseudoinverse of the matrix D (see Appendix C).Proof.
See AppendixD.Example 10.
Consider the positive system(20)xi+1=Axi+fxi+Bui,i∈N,x0=0,with(21)A=diaga1a2⋯an,ai>0fori∈σ1n,f:x~=x~1⋮x~n∈R+n↦x~12⋮x~n2=∑k=1nx~k2ek∈R+n,B=αe1,α>0,where ek is the kth column of In. The desired final state xf is assumed to be of the form xf=βe1,β>0.In this example, the mappingsG and ζ are given by(22)G:x∈R+Nn↦Gx∈R+NnGxi=∑k=1n∑j=0i-1aki-1-jxjk2ek,i∈σ1N,ζ:x∈R+Nn↦ζx∈RNnζxi=xi-∑k=1n∑j=0i-1aki-1-jxjk2ek,i∈σ1N.The applicationD is given by(23)D:u∈RN↦Du∈RNnDui=∑j=0i-1a1i-1-jαuje1,i∈σ1N.Thepseudo inverse of D is given by(24)D†:z∈RNn↦D†z=u∈RNwhere(25)u0=z11α,ui=zi+11-a1zi1α,i∈σ1N-1.Setx-=e1a1e1⋮a1N-1e1 and let P be the projection(26)P:RNn→Im+Dx↦x11e1a1x11e1⋯a1N-1x11e1T,ifx11∈R+,0,otherwise.The mappingF is given by(27)Fxi=∑k=1n∑j=0i-1aki-1-jxjk2ek+a1i-1x11e1+xN-xfx-i,i∈σ1N.In the case ofN=2, if x is a fixed point of F, then(28)x1=x11e1+x2-xfe1,x2=∑k=1nx1k2ek+a1x11e1.Hence(x1)1=(x1)1+x2-xf and (x1)i=0 for i∈σ2n. Thus x1=(x1)1e1 and xf=x2. Then the set of fixed points of F is given by(29)SF=-a1+a12+4β2e1βe1T.We have(30)ζx1=-a1+a12+4β2e1andζx2=-a12+a1a12+4β2e1,and therefore(31)U+=D†ζx∈R+2:x∈SF=-a1+a12+4β2α0T.


### 3.2. Characterization of Controllability: Second Mapping

In this subsection, we shall characterize the controllability of system (2) and the set U+ using another mapping. For this, we put(32)A=A~AN∈RnN+1×n,and we introduce the following mappings(33)G:x∈R+nN↦GxGxN∈RnN×Rn,D:u∈RNm↦DuDuN∈RnN×Rn.Also, we define the following applications(34)ξ:R+nN×R+n→RnN×Rnxy↦xy-Ax0-Gx,F:R+nN×R+n→RnN×Rnxy↦Ax0+Gx+Pξxxf+0y-xf,where P:RnN×Rn→Im+D is any projection on Im+D. Then we have the following result.Lemma 11.
Forx∈R+nN and y∈R+n, if xy∈SF, then ξxxf∈Im+D.Proof.
See AppendixE.Proposition 12.
The nonlinear system (2) is controllable in N steps if and only if for all x0,xf∈R+n,F has a fixed point.Proof.
See AppendixF.Proposition 13.
We have(35)U+=D†ξxxf+Ψ∈R+Nm:x∈S~FandΨ∈ker⁡D,with(36)S~F≔x∈R+nN:thereexistsy∈R+nsuchthatxy∈SF.Proof.
See AppendixG.


## 3.1. Characterization of Controllability: First Mapping

The aim of this subsection is to establish a necessary and sufficient condition for the controllability of system (2) based on fixed points of a function appropriately chosen. Also, we characterize the set U+ of nonnegative controls which steer the state of system (2) from an initial state x0∈R+n at i=0 to a desired final state xf∈R+n at i=N∈N+, i.e.,(12)U+=u∈R+Nm:xNu=xf,where xNu is the state of system (2) in step N corresponding to the control u.Definition 5.
The positive image of a matrixE∈Rp×q is(13)Im+E=Eu∈Rp:u∈R+q.LetP:RnN→Im+D be any projection on Im+D (i.e., any mapping P:RnN→Im+D that satisfies Pz=z if and only if z∈Im+D) and x- be any fixed element of Im+D different from zero.We define(14)ζ:x∈R+nN↦x-A~x0-Gx∈RnN,and we consider the mapping(15)F:x∈R+nN↦A~x0+Gx+Pζx+xN-xfx-∈RnN,where · is a norm on Rn. It is to be noted that the mapping F depends on the states x0 and xf.A necessary and sufficient condition for the controllability of system (2) is given by the following.Proposition 6.
The nonlinear system (2) is controllable in N steps if and only if for all x0,xf∈R+n,F has a fixed point.Proof.
See AppendixB.Remark 7.
The fixed points ofF are independent of the choice of the projection operator P and the element x-. Indeed, let P1 and P2 be two projections on Im+D and x-1 and x-2 be two any elements not equal to zero of Im+D. Let us consider the mappings(16)F1:x∈R+nN↦A~x0+Gx+P1ζx+xN-xfx-1∈RnN,and(17)F2:x∈R+nN↦A~x0+Gx+P2ζx+xN-xfx-2∈RnN.
Letx be a fixed point of F1. By proof of Proposition 6, we have xN=xf and ζx∈Im+D. Consequently P2ζx=ζx and xN-xf=0, and then(18)F2x=A~x0+Gx+ζx=x.
Hence, ifx is a fixed point of F1, then it is also a fixed point of F2.Remark 8.
Proposition6 is still true if the expression xN-xf is substituted by the term gxN, where g:Rn→R+ is any function which satisfies gz=0 if and only if z=xf.Now, a characterization of the setU+ is given by the following result.Proposition 9.
We have(19)U+=D†ζx+Ψ∈R+Nm:x∈SFandΨ∈ker⁡D,with D† being the pseudoinverse of the matrix D (see Appendix C).Proof.
See AppendixD.Example 10.
Consider the positive system(20)xi+1=Axi+fxi+Bui,i∈N,x0=0,with(21)A=diaga1a2⋯an,ai>0fori∈σ1n,f:x~=x~1⋮x~n∈R+n↦x~12⋮x~n2=∑k=1nx~k2ek∈R+n,B=αe1,α>0,where ek is the kth column of In. The desired final state xf is assumed to be of the form xf=βe1,β>0.In this example, the mappingsG and ζ are given by(22)G:x∈R+Nn↦Gx∈R+NnGxi=∑k=1n∑j=0i-1aki-1-jxjk2ek,i∈σ1N,ζ:x∈R+Nn↦ζx∈RNnζxi=xi-∑k=1n∑j=0i-1aki-1-jxjk2ek,i∈σ1N.The applicationD is given by(23)D:u∈RN↦Du∈RNnDui=∑j=0i-1a1i-1-jαuje1,i∈σ1N.Thepseudo inverse of D is given by(24)D†:z∈RNn↦D†z=u∈RNwhere(25)u0=z11α,ui=zi+11-a1zi1α,i∈σ1N-1.Setx-=e1a1e1⋮a1N-1e1 and let P be the projection(26)P:RNn→Im+Dx↦x11e1a1x11e1⋯a1N-1x11e1T,ifx11∈R+,0,otherwise.The mappingF is given by(27)Fxi=∑k=1n∑j=0i-1aki-1-jxjk2ek+a1i-1x11e1+xN-xfx-i,i∈σ1N.In the case ofN=2, if x is a fixed point of F, then(28)x1=x11e1+x2-xfe1,x2=∑k=1nx1k2ek+a1x11e1.Hence(x1)1=(x1)1+x2-xf and (x1)i=0 for i∈σ2n. Thus x1=(x1)1e1 and xf=x2. Then the set of fixed points of F is given by(29)SF=-a1+a12+4β2e1βe1T.We have(30)ζx1=-a1+a12+4β2e1andζx2=-a12+a1a12+4β2e1,and therefore(31)U+=D†ζx∈R+2:x∈SF=-a1+a12+4β2α0T.


## 3.2. Characterization of Controllability: Second Mapping

In this subsection, we shall characterize the controllability of system (2) and the set U+ using another mapping. For this, we put(32)A=A~AN∈RnN+1×n,and we introduce the following mappings(33)G:x∈R+nN↦GxGxN∈RnN×Rn,D:u∈RNm↦DuDuN∈RnN×Rn.Also, we define the following applications(34)ξ:R+nN×R+n→RnN×Rnxy↦xy-Ax0-Gx,F:R+nN×R+n→RnN×Rnxy↦Ax0+Gx+Pξxxf+0y-xf,where P:RnN×Rn→Im+D is any projection on Im+D. Then we have the following result.Lemma 11.
Forx∈R+nN and y∈R+n, if xy∈SF, then ξxxf∈Im+D.Proof.
See AppendixE.Proposition 12.
The nonlinear system (2) is controllable in N steps if and only if for all x0,xf∈R+n,F has a fixed point.Proof.
See AppendixF.Proposition 13.
We have(35)U+=D†ξxxf+Ψ∈R+Nm:x∈S~FandΨ∈ker⁡D,with(36)S~F≔x∈R+nN:thereexistsy∈R+nsuchthatxy∈SF.Proof.
See AppendixG.


## 4. Observability of Nonlinear Positive Discrete Systems

As the corresponding notion of controllability, observability is obviously an important concept. In this section we discuss the problem of observability for nonlinear positive discrete systems.Consider the nonlinear systems (2)-(3) with ui=0, i∈N, and x0 is assumed to be unknown.System (2)-(3) is said to be observable in N∈N+ steps if the information of the output sequence yi∈R+r for i∈σ0N-1 is sufficient to determine uniquely the nonnegative initial state x0∈R+n.In this section, we will use the notationsx=x0x1⋮xN-1∈RnN and y=y0y1⋮yN-1∈RrN, with N∈N+ and xi∈Rn, yi∈Rr for i∈σ0N-1.To define observability more precisely, let(37)Γ:R+n→R+rNx0↦ywhere y is the output over N steps.Definition 14.
System (2)-(3) is said to be observable in N∈N+ steps if Γ is injective.The state of (2) is given by(38)xi=Aix0+∑j=0i-1Ai-1-jfxj,i∈N+.We put(39)A~=InA⋮AN-1∈RnN×n,and(40)S=C0⋯⋯00C⋱⋮⋮⋱⋱⋱⋮⋮⋱⋱00⋯⋯0C∈R+rN×nN.Let us consider the nonlinear mappingG defined by(41)G:x∈R+nN↦Gx∈RnN,with Gxi=∑j=0i-1Ai-1-jfxj, i∈σ0N-1.Then, solution (38) over N steps can be rewritten as(42)x=A~x0+Gx,and the output has the form(43)y=Sx=SA~x0+SGx.


### 4.1. Necessary and Sufficient Criteria for Observability

The goal of this subsection is to give a characterization of the setO+ of states of system (2) such that yg=Sx where yg is the given output over N steps, i.e.,(44)O+=x∈R+nN:x=A~x0+Gxandyg=Sx,and consequently we shall establish a necessary and sufficient condition for the observability of system (2)-(3).LetP:RnN→Im+A~ be any projection on Im+A~ and x- be any fixed element of Im+A~ different from zero.We define(45)ζ:x∈R+nN↦x-Gx∈RnN,and we consider the mapping(46)H:x∈R+nN↦Gx+Pζx+Sx-ygx-∈RnN.The coming result gives a characterization of the setO+.Proposition 15.
Letx∈R+nN. Then x is an element of O+ if and only if x is a fixed point of H.Proof.
It is similar to that of Proposition6.Example 16.
Consider the positive system(47)xi+1=Axi+fxi,i∈N,x0∈R+n,yi=Cxi,with(48)A=diaga1a2⋯an,ai>0fori∈σ1n,C=10⋯0,and(49)f:x~=x~1⋮x~n∈R+n↦x~120⋮0=x~12e1∈R+n.
The mappingsG and ζ for this example are given by(50)G:x∈R+nN↦Gx∈R+nNGxi=∑j=0i-1a1i-1-jxj12e1,i∈σ0N-1,ζ:x∈R+nN↦ζx∈RnNζxi=xi-∑j=0i-1a1i-1-jxj12e1,i∈σ0N-1.
Setx-=e1a1e1⋮a1N-1e1 and introduce the projection P(51)P:RnN→Im+A~x↦x0TAx0T⋯AN-1x0TT,ifx0∈R+n,0,otherwise.
The mappingH is given by(52)Hxi=∑j=0i-1a1i-1-jxj12e1+∑j=1najix0jej+Sx-ygx-i,i∈σ0N-1.ForN=2 and yg=11+a1T, the set of fixed points of H, and hence O+, are (53)O+=1x02⋯x0n1+a1a2x02⋯anx0nT:x0i∈R+,i∈σ2n.The following proposition gives a necessary and sufficient condition for the observability of our system.Proposition 17.
System (2)-(3) is observable in N steps if and only if for every given output yg∈R+rN, H has at most one fixed point.Proof.
See AppendixH.


### 4.2. Another Characterization of the Observability

The aim of this subsection is to give a second characterization of the setO+ and of the observability of system (2)-(3) based on the fixed points of another function appropriately chosen.We pose(54)A=A~SA~∈RNr+n×n,and we introduce the following mappings(55)G:R+nN→RnN×RrNx↦GxSGx,ξ:R+nN×R+rN→RnN×RrNxz↦xz-Gx.We define(56)H:R+nN×R+rN→RnN×RrNxz↦Gx+Pξxyg+0z-ygwhere P:RnN×RrN→Im+A is any projection on Im+A.Lemma 18.
Forx∈R+nN and z∈R+rN, if xz∈SH, then ξxyg∈Im+A.Proof.
It is similar to that of Lemma11.Proposition 19.
The setO+ is given by(57)O+=S~H≔x∈R+nN:thereexistsz∈R+rNsuchthatxz∈SH.Proof.
It is similar to that of Proposition12.Proposition 20.
System (2)-(3) is observable in N steps if and only if for every given output yg∈R+rN, H has at most one fixed point.Proof.
It is similar to that of Proposition17.


## 4.1. Necessary and Sufficient Criteria for Observability

The goal of this subsection is to give a characterization of the setO+ of states of system (2) such that yg=Sx where yg is the given output over N steps, i.e.,(44)O+=x∈R+nN:x=A~x0+Gxandyg=Sx,and consequently we shall establish a necessary and sufficient condition for the observability of system (2)-(3).LetP:RnN→Im+A~ be any projection on Im+A~ and x- be any fixed element of Im+A~ different from zero.We define(45)ζ:x∈R+nN↦x-Gx∈RnN,and we consider the mapping(46)H:x∈R+nN↦Gx+Pζx+Sx-ygx-∈RnN.The coming result gives a characterization of the setO+.Proposition 15.
Letx∈R+nN. Then x is an element of O+ if and only if x is a fixed point of H.Proof.
It is similar to that of Proposition6.Example 16.
Consider the positive system(47)xi+1=Axi+fxi,i∈N,x0∈R+n,yi=Cxi,with(48)A=diaga1a2⋯an,ai>0fori∈σ1n,C=10⋯0,and(49)f:x~=x~1⋮x~n∈R+n↦x~120⋮0=x~12e1∈R+n.
The mappingsG and ζ for this example are given by(50)G:x∈R+nN↦Gx∈R+nNGxi=∑j=0i-1a1i-1-jxj12e1,i∈σ0N-1,ζ:x∈R+nN↦ζx∈RnNζxi=xi-∑j=0i-1a1i-1-jxj12e1,i∈σ0N-1.
Setx-=e1a1e1⋮a1N-1e1 and introduce the projection P(51)P:RnN→Im+A~x↦x0TAx0T⋯AN-1x0TT,ifx0∈R+n,0,otherwise.
The mappingH is given by(52)Hxi=∑j=0i-1a1i-1-jxj12e1+∑j=1najix0jej+Sx-ygx-i,i∈σ0N-1.ForN=2 and yg=11+a1T, the set of fixed points of H, and hence O+, are (53)O+=1x02⋯x0n1+a1a2x02⋯anx0nT:x0i∈R+,i∈σ2n.The following proposition gives a necessary and sufficient condition for the observability of our system.Proposition 17.
System (2)-(3) is observable in N steps if and only if for every given output yg∈R+rN, H has at most one fixed point.Proof.
See AppendixH.


## 4.2. Another Characterization of the Observability

The aim of this subsection is to give a second characterization of the setO+ and of the observability of system (2)-(3) based on the fixed points of another function appropriately chosen.We pose(54)A=A~SA~∈RNr+n×n,and we introduce the following mappings(55)G:R+nN→RnN×RrNx↦GxSGx,ξ:R+nN×R+rN→RnN×RrNxz↦xz-Gx.We define(56)H:R+nN×R+rN→RnN×RrNxz↦Gx+Pξxyg+0z-ygwhere P:RnN×RrN→Im+A is any projection on Im+A.Lemma 18.
Forx∈R+nN and z∈R+rN, if xz∈SH, then ξxyg∈Im+A.Proof.
It is similar to that of Lemma11.Proposition 19.
The setO+ is given by(57)O+=S~H≔x∈R+nN:thereexistsz∈R+rNsuchthatxz∈SH.Proof.
It is similar to that of Proposition12.Proposition 20.
System (2)-(3) is observable in N steps if and only if for every given output yg∈R+rN, H has at most one fixed point.Proof.
It is similar to that of Proposition17.


## 5. Conclusion

The fixed point technique is an important tool used in mathematics to treat different nonlinear problems [24–26]. In this work we have employed this tool for resolving the controllability and observability problem for nonlinear positive discrete systems. Necessary and sufficient conditions for the positivity of our discrete system have been established (Proposition 3). Criteria for the controllability (Propositions 6 and 12) and observability (Propositions 17 and 20) have been proved. A characterization of nonnegative controls which drives the state of the system from its initial value to a given desired final state is given (Propositions 9 and 13). The set of all nonnegative states which correspond to the given output is also characterized (Propositions 15 and 19). In our future work, we investigate the controllability and observability of positive nonlinear continuous systems.



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*Source: 3279290-2018-12-03.xml*