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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 3279290, 9 pages
https://doi.org/10.1155/2018/3279290
Research Article

Controllability and Observability Analysis of Nonlinear Positive Discrete Systems

1Laboratory of Analysis, Modeling and Simulation, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco
2Laboratory of Information Technology and Modeling, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco

Correspondence should be addressed to Mouhcine Naim; moc.liamg@3102enichuommian

Received 18 September 2018; Accepted 18 November 2018; Published 3 December 2018

Academic Editor: Carmen Coll

Copyright © 2018 Mouhcine Naim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [1219]. 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. is the set of nonnegative integers, the set of positive integers, the finite subset of with , the set of real vectors with components, the set of all vectors in with nonnegative components, i.e.,where denotes the transpose, the th component of a vector , the set of real constant matrices of size , and the identity matrix in . In addition, if is a mapping, then denotes the set of all its fixed points.

In this work, we consider the discrete nonlinear system described by

Information on system (2) is given by the output equationwhere is the system state, is the input (or control), , , , , is a nonlinear function, and 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 and all nonnegative inputs , , we have and for any , where is the solution corresponding to and , and, similarly, is the output corresponding to and

Definition 2. A matrix is said to be nonnegative and denoted by , if all of its elements are nonnegative, i.e., for all

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.

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 steps if for any initial state and any desired final state , there exists an input sequence , which steers the state of the system from to , i.e., .

The explicit solution of (2) is given by

In this section, we will use the following notation: , with , for and for

We define the matrix

Let us consider the nonlinear mapping defined bywithand let denote the linear mapping defined bywith

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 of nonnegative controls which steer the state of system (2) from an initial state at to a desired final state at , i.e.,where is the state of system (2) in step corresponding to the control

Definition 5. The positive image of a matrix is

Let be any projection on (i.e., any mapping that satisfies if and only if and be any fixed element of different from zero.

We defineand we consider the mappingwhere is a norm on . It is to be noted that the mapping depends on the states and

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 steps if and only if for all has a fixed point.

Proof. See Appendix B.

Remark 7. The fixed points of are independent of the choice of the projection operator and the element Indeed, let and be two projections on and and be two any elements not equal to zero of . Let us consider the mappingsandLet be a fixed point of . By proof of Proposition 6, we have and . Consequently and , and thenHence, if is a fixed point of , then it is also a fixed point of

Remark 8. Proposition 6 is still true if the expression is substituted by the term , where is any function which satisfies if and only if

Now, a characterization of the set is given by the following result.

Proposition 9. We havewith being the pseudoinverse of the matrix (see Appendix C).

Proof. See Appendix D.

Example 10. Consider the positive systemwithwhere is the th column of . The desired final state is assumed to be of the form

In this example, the mappings and are given by

The application is given by

The pseudo inverse of is given bywhere

Set and let be the projection

The mapping is given by

In the case of , if is a fixed point of , then

Hence and for . Thus and Then the set of fixed points of is given by

We haveand therefore

3.2. Characterization of Controllability: Second Mapping

In this subsection, we shall characterize the controllability of system (2) and the set using another mapping. For this, we putand we introduce the following mappings

Also, we define the following applicationswhere is any projection on Then we have the following result.

Lemma 11. For and , if , then

Proof. See Appendix E.

Proposition 12. The nonlinear system (2) is controllable in steps if and only if for all has a fixed point.

Proof. See Appendix F.

Proposition 13. We havewith

Proof. See Appendix G.

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 , , and is assumed to be unknown.

System (2)-(3) is said to be observable in steps if the information of the output sequence for is sufficient to determine uniquely the nonnegative initial state

In this section, we will use the notations and , with and , for

To define observability more precisely, letwhere is the output over steps.

Definition 14. System (2)-(3) is said to be observable in steps if is injective.

The state of (2) is given by

We putand

Let us consider the nonlinear mapping defined bywith ,

Then, solution (38) over steps can be rewritten asand the output has the form

4.1. Necessary and Sufficient Criteria for Observability

The goal of this subsection is to give a characterization of the set of states of system (2) such that where 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 be any projection on and be any fixed element of different from zero.

We defineand we consider the mapping

The coming result gives a characterization of the set .

Proposition 15. Let Then is an element of if and only if is a fixed point of

Proof. It is similar to that of Proposition 6.

Example 16. Consider the positive systemwithandThe mappings and for this example are given bySet and introduce the projection The mapping is given by

For and , the set of fixed points of , and hence , are

The following proposition gives a necessary and sufficient condition for the observability of our system.

Proposition 17. System (2)-(3) is observable in steps if and only if for every given output , has at most one fixed point.

Proof. See Appendix H.

4.2. Another Characterization of the Observability

The aim of this subsection is to give a second characterization of the set and of the observability of system (2)-(3) based on the fixed points of another function appropriately chosen.

We poseand we introduce the following mappings

We definewhere is any projection on .

Lemma 18. For and , if , then

Proof. It is similar to that of Lemma 11.

Proposition 19. The set is given by

Proof. It is similar to that of Proposition 12.

Proposition 20. System (2)-(3) is observable in steps if and only if for every given output , has at most one fixed point.

Proof. It is similar to that of Proposition 17.

5. Conclusion

The fixed point technique is an important tool used in mathematics to treat different nonlinear problems [2426]. 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.

Appendix

A.

Proof. (Sufficiency) From (2), for , we havesince (4) holds, and Similarly, for , we obtain since (4) and (A.1) hold and Repeating the procedure for , we have for every . Consequently, if , then for any . Thus system (2)-(3) is positive.
(Necessity) If system (2)-(3) is positive, then, in particular for , we have and . Then since is arbitrary.
Suppose . Hence , and consequently . Now assuming that , then . Suppose one of the components of , , is negative. Then, for the nonnegative vector with being the th component, the th component of would be . But, if , then , and this is absurd, which completes the proof.

B.

Proof. (Sufficiency) Let , If is a fixed point of , thenHencewhich implies that and , which ensures that because
Since , then there exists an input such thatConsequently, (B.1) becomesThusHence system (2) is controllable in steps.
(Necessity) Let , . Since system (2) is controllable in steps, there exists an input such that . Then we havewith being the solution of system (2) over steps corresponding to the control .
Consequentlyand thenHence, we obtainThus is a fixed point of the mapping . The proposition is proved.

C.

Definition C.1 (see [27]). A matrix is said to be the Moore-Penrose generalized inverse (pseudo inverse) of if(i),(ii),(iii),(iv)

Lemma C.2 (see [28]). For any matrix , its Moore-Penrose generalized inverse matrix is existent and unique.

D.

Proof. If , then by proof of Proposition 6, the trajectory of system (2) corresponding to control is a fixed point of and . Moreover, we can writewith and, by Definition C.1, we havei.e.,
Conversely, let with and ; thenSince is a fixed point of , then by proof of Proposition 6, we have and . Consequently there exists an input such thatThusHence ; thus . This finishes the proof.

E.

Proof. Let and . If , thenand hencewhich implies that .

F.

Proof. (Sufficiency) Let , If , then by Lemma 11, we have , so there exists an input such thatOn the other hand, by (34) we havewhich implies that(Necessity) Let , . Since the system (2) is controllable in steps, there exists an input such that
Hencewhere is the solution of system (2) corresponding to the control .
Thenwhich implies that , and consequentlyTherefore, we haveand hence . This completes the proof.

G.

Proof. If , then by proof of Proposition 12, we have and , with the solution of system (2) corresponding to the control . Moreover, we can writewith and we havei.e., .
Conversely, let with and ; thenand by the proof of Proposition 12, we have and . Consequently there exists an input such thatthenHence ; thus . The proposition is proved.

H.

Proof. System (2)-(3) is observable in steps if and only if, for all , there exists at most one such thatwhere is the trajectory of system (2) corresponding to the initial state . Consequently, system (2)-(3) is observable if and only if the set , and hence , contains at most one element.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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