Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

Abstract

Abstract. Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows for the possibility of rotating between different courses of management action. In developing our control strategies we appeal to comparison and monotonicity arguments. Interestingly, componentwise nonnegativity of the model, combined with an irreducibility assumption, implies that several issues typically associated with high-gain adaptive controllers do not arise and usual high-gain structural assumptions are not required.

Adaptive stabilization of impulsive switched linear time-delay systems: A piecewise dynamic gain approach

“…Remark 3 Some comments are needed to clarify that the family of Riccati equations ( 10) can be introduced without loss of generality. Since (A p , B p ) is controllable for all p ∈ M , one can always find a solution for P p and Q p satisfying (10). As a matter of fact, the Riccati equations guarantee a sufficient large stability margin with the only requirement of controllability.…”

“…Thanks to their capability of modeling a wide range of systems with interacting continuous/discrete dynamics [1,2], hybrid systems with impulsive and switching dynamics, usually called impulsive switched systems, have been attracting fruitful lines of research, encompassing stability, stabilization [3,4,5], robust control [6,7], and others. Dealing effectively with large parametric uncertainty is becoming increasingly crucial when controlling hybrid systems: in several applications it has been recognized that robust controllers may give rise to rather conservative performance in the presence of a large and nonpolytopic uncertainty set [8,9,10]. Therefore, the design of adaptive control methods to cope with large and nonpolytopic parametric uncertainties in hybrid systems is often relevant.…”

A switching feedback control approach for persistence of managed resources

“…We note that in natural resource management the word "adaptive" generally means a feedback, see [36]. Under the assumption that at least one of these strategies is stabilizing, and by carefully exploiting the rules by which switching is determined, in [18] we were able to demonstrate convergence of the scheme with switching terminating at a strategy that was itself stabilizing. In developing this approach, much use was made of the underlying positive systems structure, that is, dynamical systems whose evolution map leaves a positive cone invariant; see, for instance [4,5].…”

“…We design an algorithm for switching between strategies which identifies (or learns) a suitable strategy that ensures persistence. To give an outline of our approach, we highlight our previous work [18] which addressed the problem of eradication of pests using a so-called adaptive feedback control scheme, where the feedback switches through a number of distinct control strategies. Adaptive control is a broad term, with no one single agreed definition, and traces its roots back to the control of aircraft in the 1950s.…”

On the strict monotonicity of spectral radii for classes of bounded positive linear operators

“…The assumptions in (1.4) and (1.5) are too conservative for many applications, however. We are interested in the strict inequality (1.1) owing to its utility for discrete-time positive dynamical systems, where comparison arguments are readily applicable, such as [10], particularly in the infinite-dimensional case. For example, much attention has been devoted in theoretical ecology to discrete-time dynamical systems specified by certain classes of integral operators, so-called Integral Projection Models [7,8].…”