Analysis and Synthesis of Interconnected Positive Systems

Abstract

This paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix. We first show that admissibility, to be defined in this paper, is an essential requirement in constructing such interconnected systems. Then, we clarify that the interconnected system is admissible and stable if and only if a Metzler matrix, which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix, is Hurwitz stable. By means of this key result, we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the Frobenius eigenvalue of the interconnection matrix and the weighted L 1-induced norm of the positive subsystems again to be defined in this paper. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., its state converges to a unique strictly positive vector (that is known in advance up to a strictly positive constant multiplicative factor) for any nonnegative and nonzero initial state. As an important consequence of this property, we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its Frobenius eigenvalue, where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady-stage gains of the positive subsystems. This result is then naturally and effectively applied to formation control of multi-agent systems with positive dynamics. This result can be seen as a generalization of a well-known consensus algorithm that has been basically applied to interconnected systems constructed from integrators.

Analysis and Synthesis of Discrete-Time Interconnected Positive Systems

“…The interconnected system G Ω trivially satisfies the admissibility in the case where D = 0 while the admissibility is a sufficient condition for the positivity of G Ω in the case where D 0. These features are similar to those for the continuous-time setting ([5], [6]). …”

“…The definition of the admissibility is similar to that for continuous-time interconnected positive systems ( [5], [6]). If det(I − DΩ) 0, then the interconnection is well-posed and the state-space realization of G Ω is represented by…”

“…, [6]) Consider the case where the subsystem G i (i ∈ Z N ) is SISO. Then, for given Ω ∈ R N×N , the interconnected systemG Ω is admissible and stable if and only if ρ(ΨΩ) < 1, where…”

“…The parametrization of the interconnection matrices satisfying conditions (iv) and (v) in Theorem 3.1 is available ( [5], [6]) for a given shape of formation. For example, consider the case where γ = 1 and the communication topology among subsystems is a ring.…”

“…One direction to deal with large-scale systems is interconnected positive systems ( [5], [6]) where the plant is a gathering of positive subsystems while the controller is a nonnegative interconnection matrix. If an interconnected positive system has so-called admissibility, then the system is well-posed, and the positivity of the subsystem outputs is preserved.…”