Positive Input Reachability And Controllability Of Positive Systems

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Linear Algebra And Its Applications


We study controllability and reachability for discrete linear control systems, x(k + 1) = Ax(k) + Bu(k), in which the state vector x(k), the control vector u(k), the n × n matrix A, and the n × m matrix B have nonnegative entries. For an unconstrained system, controllability is equivalent to reachability from the zero state in n steps. Furthermore, (A, B) is controllable if and only if the matrix Cₙ = [B AB A²B ⋅⋅⋅ Aⁿ⁻¹B] has rank n. We show that these properties are not equivalent for positive systems. This has important implications for control strategy. The timing of control inputs and the size and structure of the control matrix B are much more critical. The zero-nonzero pattern of Cₙ plays a crucial role. We illustrate these fundamental differences with models of a two species reversible chemical reaction, and with two pharmacokinetic models of drug distribution in the human body.