Controllability and Observability Analysis of Nonlinear Positive Discrete Systems

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.


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][13][14][15][16][17][18][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.

Preliminaries
First we introduce some notations.N is the set of nonnegative integers, N + the set of positive integers,  푘 푠 = {,  + 1, . . ., } the finite subset of N with  ≤ , R 푛 the set of real vectors with  components, R 푛 + the set of all vectors in R 푛 with nonnegative components, i.e., where  denotes the transpose, ( 푖 ) 푗 the th component of a vector  푖 , R 푛×푚 the set of real constant matrices of size  × , and  푛 the identity matrix in R 푛×푛 .In addition, if  is a mapping, then  퐹 denotes the set of all its fixed points.
In this work, we consider the discrete nonlinear system described by (2) Information on system (2) is given by the output equation where 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
Proof.See Appendix A.
In the remainder of this paper, we assume that system (2)-( 3) is positive.

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 + steps if for any initial state  0 ∈ R 푛 + and any desired final state , which steers the state of the system from  0 to  푓 , i.e.,  푁 =  푓 .
The explicit solution of ( 2) is given by In this section, we will use the following notation:  = ( Let us consider the nonlinear mapping  defined by with and let  denote the linear mapping defined by with Then, solution (5) over  steps can be rewritten as 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 where  푁 () is the state of system (2) in step  corresponding to the control .
Definition 5.The positive image of a matrix  ∈ R 푝×푞 is Let  : R 푛푁 → Im +  be any projection on Im +  (i.e., any mapping  : R 푛푁 → Im +  that satisfies  =  if and only if  ∈ Im + ) and  be any fixed element of Im +  different from zero.
We define and we consider the mapping where ‖ ⋅ ‖ is a norm on R 푛 .It is to be noted that the mapping  depends on the states  0 and  푓 .
A necessary and sufficient condition for the controllability of system (2) is given by the following.Proposition 6.The nonlinear system (2) and Let  be a fixed point of  1 .By proof of Proposition 6, we have  푁 =  푓 and () ∈ Im + .Consequently  2 () = () and ‖ 푁 −  푓 ‖ = 0, and then Hence, if  is a fixed point of  1 , then it is also a fixed point of  2 .
Remark 8. Proposition 6 is still true if the expression ‖ 푁 −  푓 ‖ is substituted by the term ( 푁 ), where  : R 푛 → R + is any function which satisfies () = 0 if and only if  =  푓 .Now, a characterization of the set U + is given by the following result.
Proposition 9. We have with  † being the pseudoinverse of the matrix  (see Appendix C).
Proof.See Appendix D.
Example 10.Consider the positive system with where  푘 is the th column of  푛 .The desired final state  푓 is assumed to be of the form  푓 =  1 ,  > 0.
In this example, the mappings  and  are given by The application  is given by The pseudo inverse of  is given by where Set  = ( ) and let  be the projection The mapping  is given by In the case of  = 2, if  is a fixed point of , then Hence ( 1 ) 1 = ( 1 ) 1 + ‖ 2 −  푓 ‖ and ( 1 ) 푖 = 0 for  ∈  푛 2 .Thus  1 = ( 1 ) 1  1 and  푓 =  2 .Then the set of fixed points of  is given by We have and therefore

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 and we introduce the following mappings Also, we define the following applications where P : R 푛푁 × R 푛 → Im + D is any projection on Im + D.
Then we have the following result.

Lemma 11. For
Proposition 12.The nonlinear system (2) is controllable in  steps if and only if for all  0 ,  푓 ∈ R 푛 + , F has a fixed point.
Proof.See Appendix F.
Proposition 13.We have with Proof.See Appendix G.

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  푖 = 0,  ∈ N, and  0 is assumed to be unknown.System (2)-( 3) is said to be observable in  ∈ N + steps if the information of the output sequence  푖 ∈ R 푟 + for  ∈  푁−1 0 is sufficient to determine uniquely the nonnegative initial state  0 ∈ R 푛 + .
In this section, we will use the notations  = ( To define observability more precisely, let where  is the output over  steps. Definition 14. System (2)-( 3) is said to be observable in  ∈ N + steps if Γ is injective.
The state of ( 2) is given by We put and Let us consider the nonlinear mapping  defined by with . Then, solution (38) over  steps can be rewritten as and the output has the form  =  =  Ã 0 +  () . (43) 4.1.Necessary and Sufficient Criteria for Observability.The goal of this subsection is to give a characterization of the set O + of states of system (2) such that  푔 =  where  푔 is the given output over  steps, i.e., and consequently we shall establish a necessary and sufficient condition for the observability of system (2)-(3).Let  : R 푛푁 → Im + Ã be any projection on Im + Ã and  be any fixed element of Im + Ã different from zero.
We define and we consider the mapping The coming result gives a characterization of the set O + .
Proof.It is similar to that of Proposition 6.

Example 16. Consider the positive system
with and The mappings  and  for this example are given by Set  = ( ) and introduce the projection   : The mapping  is given by For  = 2 and  푔 = (1 1 +  1 ) 푇 , the set of fixed points of , and hence O + , are The following proposition gives a necessary and sufficient condition for the observability of our system.We pose and we introduce the following mappings We define where P : R 푛푁 × R 푟푁 → Im + A is any projection on Im + A.

Conclusion
The fixed point technique is an important tool used in mathematics to treat different nonlinear problems [24][25][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.Hence  = Ã 0 + () + ; thus  ∈ U + .This finishes the proof.
and hence

G.
Proof.If  ∈ U + , then by proof of Proposition 12, we have (

H.
Proof.System (2)-( 3) is observable in  steps if and only if, for all  푔 ∈ R 푟푁 + , there exists at most one  0 ∈ R 푛 + such that  = Ã 0 +  () , where  is the trajectory of system (2) corresponding to the initial state  0 .Consequently, system (2)-( 3) is observable if and only if the set O + , and hence  퐻 , contains at most one element.

4. 2 .
Another Characterization of the Observability.The aim of this subsection is to give a second characterization of the set O + and of the observability of system (2)-(3) based on the fixed points of another function appropriately chosen.
∈ N, we have  푖 ∈ R 푛 + and  푖 ∈ R 푟 + for any  ∈ N, where  푖 is the solution corresponding to  0 and  푖 , and, similarly,  푖 is the output corresponding to  0 and  푖 .
is controllable in  steps if and only if for all  0 ,  푓 ∈ R 푛 + ,  has a fixed point.The fixed points of  are independent of the choice of the projection operator  and the element .Indeed, let  1 and  2 be two projections on Im +  and  1 and  2 be two any elements not equal to zero of Im + .Let us consider the mappings

Proposition 17 .
System (2)-(3) is observable in  steps if and only if for every given output  푔 ∈ R 푟푁 + ,  has at most one fixed point.
Proof.It is similar to that of Lemma 11.It is similar to that of Proposition 17.