Positive state controllability of positive linear systems
Introduction
Controllability is one of the most fundamental concepts in control theory. For finite-dimensional, linear, time-invariant, continuous time systems the notions of reachability, controllability and null controllability are all equivalent. The formulation of these concepts dates back to Kalman [1] and, as is well-known, their appeal lies in the interplay between analytic and algebraic concepts. For instance, the existence of a control steering the system to a desired state is equivalent to the reachability matrix having full rank.
Controllability does not a priori respect any (componentwise) nonnegativity of a system. This is problematic for many physically motivated applications, where state and input variables correspond to quantities that cannot take negative values. The need for nonnegative variables motivated the development of positive system theory and there now exist several textbooks on the subject (for example, [2], [3], [4]). Naturally, controllability, that is positive input controllability, in such a framework is more limited than the general case, but the situation is well understood ([5], [6] and the references therein). A key feature of positive system theory is the notion that both the state and the input variables must be nonnegative.
It is of interest, however, to consider where and are componentwise nonnegative under the constraints that just the state must be nonnegative—what might be termed positive state controllability. There are conceivably many applications of such a framework, for example, in economic or logistic type models (see, for example, Miller and Blair [7]). Our primary example of where such a framework is necessary, however, is population ecology. Here matrix models are often used (see, for example, Caswell [8] or Cushing [9]) with the nonnegative state denoting a stage- or age-structured population, and the control denoting a conservation strategy or a form of pest control or harvesting. There are many papers (including, for example, [10], [11], [12], [13]) where the model (1.1) is suitable for describing the addition or removal of individuals from a population and for a full description of these actions we require that can take negative values.
The framework of positive state controllability places a nonnegativity constraint on the codomain, and not on the domain, of the input-to-state map and it is not immediately clear that the positive input controllability theory is applicable. Here we demonstrate that under certain assumptions (reasonable for applications to population ecology) the problem of positive state controllability is equivalent to positive input controllability of a related positive system. Using this approach we characterise both the set of reachable states and the set of null controllable states of the pair under the constraint that the state must remain nonnegative. We demonstrate that, for example, the class of Leslie matrices [14] (with suitable control) that is frequently used in ecological modelling is positive state controllable, but often with negative control signals. We believe that there is seemingly a non-trivial ‘middle ground’ between the controllability of linear systems and the positive input controllability of positive systems that is worthy of in-depth study.
Section snippets
Positive state control
For denotes the nonnegative orthant in and is the th standard basis vector. For vectors and matrices (also ) and (also ) denotes componentwise nonnegativity. The superscript denotes matrix transposition. We are interested in the pair generating the controlled system (1.1) where and the state is nonnegative.
Our main result is Theorem 2.6 which relates nonnegative state trajectories with possibly nonpositive inputs to nonnegative state
Examples
One motivation for studying positive state controllability of the linear system (1.1) is the class of systems that arise in population ecology. Here we present some examples. Example 3.1 We recall that an Leslie [14] matrix has the following structure which models a population partitioned into discrete, increasing age-stages. Correspondingly, denote reproductive rates and denote survival rates, the latter as proportions are each no greater than one. For
Acknowledgements
Chris Guiver is fully supported and Dave Hodgson and Stuart Townley are partially supported by EPSRC grant EP/I019456/1.
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