On the reachability in any fixed time for positive continuous-time linear systems
Introduction
Positive linear systems appear naturally in the modeling of a lot of practical problems in economics, engineering and biology [8], [1]. The reachability of such systems has been studied in the past but mainly for discrete-time systems [5], [6], [10], see [2], [3], [4] for more recent references in which it appears that this problem is now well solved and understood.
It seems that the reachability of continuous-time systems is harder and was much less studied, see [9] for a classical reference and [7] for a more recent state of the art. A common (and surprising) feature of these works is that they use as a definition of reachability the possibility of steering the state in any point of the positive orthant in finite time. In this paper we consider the possibility of steering the state in any point of the positive orthant in any fixed time. This property is referred to in this paper as the strong reachability. This seems to us to be more in the spirit of the original definition of reachability although the two definitions coincide in the case where the input function is not restricted to be nonnegative. In the case of positive systems, however, the two definitions are not equivalent. The example 13 of [7] shows a system that is reachable in finite time but not reachable in any fixed time (for some parameters choice).
The main result of this paper is essentially that the only strongly reachable positive systems are those made of decoupled scalar subsystems. For nonstrongly reachable positive systems, a complete characterization of the set of strongly reachable states is given.
The paper is organized as follows. In Section 2 we review some well-known material on positive systems and their graph representation. In Section 3 we prove that the scalar systems are strongly reachable and give the necessary and sufficient conditions for strong reachability. Section 4 provides the strongly reachable set.
Section snippets
Linear positive systems: definitions and basic results
In this paper we denote by the set of nonnegative real numbers. A matrix L is said to be positive if for , and there exists indices such that .
We study continuous-time linear time-invariant systems of the following form:where the matrices A and B are real valued matrices of respective dimensions and . The matrix B is a positive matrix and A is of Metzler type that is, for and . The initial state
Scalar positive systems are strongly reachable
We consider a scalar positive system as in (1) with one input and one state ():where .
Denote by the unit step function. With and we get for system (2) the response for and for . Then, for any and any , using the input functionandwe get and the system is therefore strongly reachable. This proves the following. Proposition 2 The scalar system defined in (2) is strongly
The strongly reachable set
In this section we will use the following notations. denotes the Euclidian norm, and denote, respectively, the interior and the closure of a given set in the induced topology.
Define the strongly reachable set as for and any there exists a nonnegative piecewise continuous function such that .
A first property of the strongly reachable set is easily obtained. Proposition 4 is a convex cone. Proof Follows directly from the linearity and the positivity of the system. □
Conclusion
In this paper the problem of reachability for continuous-time positive systems is studied in the case where we ask for this reachability in any fixed time. The condition for such a reachability are given together with the set of states which are reachable in this sense.
The results of this paper may appear to be “intuitively obvious” since reachability in any fixed time induces input functions which are approximates of impulse distributions. Since only the Dirac distribution can be approached by
References (10)
- et al.
Canonical forms for positive discrete-time linear control systems
Linear Algebra Appl.
(2000) A simple graph theoretic characterization of reachability for positive linear systems
Systems Control Lett.
(2004)- et al.
Positive input reachability and controllability of positive linear systems
Linear Algebra Appl.
(1987) - L. Benvenuti, A. de Santis, L. Farina, Positive systems, in: Proceedings of the Symposium POSTA 2003, Lecture Notes in...
- et al.
A survey of reachability and controllability for positive systems
Ann. Oper. Res.
(2000)
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2014, AutomaticaCitation Excerpt :These systems are of great practical importance, as the nonnegative property occurs quite frequently in industrial applications, biological systems, human interactions, and in nature, e.g. the control of anesthetic or insulin (Haddad, Hayakawa, & Bailey, 2003; Luni & Doyle, 2011; Syafiie, Chong, & Chang, 2011) within the human body. Various properties of positive systems and related systems have been studied over the years, e.g. results on positive system reachability, switched systems observer design, LMI and linear programming approaches to positive stability, positive observers, 2D systems, general stability/output feedback and optimal control designs can be found in Back and Astolfi (2008), Beauthier and Winkin (2010), Commault and Alamir (2007), Ebihara, Peaucelle, and Arzelier (2012), Kaczorek (2009), Li and Lam (2012), Middleton, Colaneri, Hernandez-Vargas, and Blanchini (2010), Rami, Tadeo, and Helmke (2011), Roszak and Davison (2009a), Shu, Lam, Gao, Du, and Wu (2008), Sun and Ge (2005), Zhu, Meng, and Zhang (2013), and references therein. We also refer the interested reader to Farina and Rinaldi (2000) and Kaczorek (2002) for an in-depth reference guide to positive systems.
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