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Abstract-To simplify the analysis of complex dynamical networks, we have recently proposed an approach that decomposes the overall system into the sign-definite interconnection of subsystems with a Positive Impulse Response (PIR). PIR systems include and significantly generalise input-output monotone systems, and the PIR property (or equivalently, for linear systems, the Monotonic Step Response property) can be evinced from experimental data, without an explicit model of the system. An aggregate of PIR subsystems can be associated with a signed matrix of interaction weights, hence with a signed graph where the nodes represent the subsystems and the arcs represent the interactions among them. In this paper, we prove that stability is structurally ensured (for any choice of the PIR subsystems) if a Metzler matrix depending on the interaction weights is Hurwitz; this condition is non-conservative. We also show how to compute an influence matrix that represents the steady-state effects of the interactions among PIR subsystems.

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Abstract-We have recently considered the problem of tuning a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope of matrices: to drive the output to a given desired value, we have suggested an integral feedback scheme, whose convergence is ensured if the polytope of matrices is robustly full row rank. The suggested tuning scheme may fail in the presence of parasitic dynamics, which may destabilize the loop if the tuning action is too aggressive. Here we show that such tuning action can be applied to dynamic plants as well if it is sufficiently "slow", a property that we can ensure by limiting the integral action. We provide robust bounds based on the exclusive knowledge of the largest time constant and of the matrix polytope to which the system Jacobian is known to belong. We also provide similar bounds in the presence of parasitic dynamics affecting the actuators.

Merging two Control Lyapunov Functions (CLFs) means creating a single "new-born" CLF by starting from two parents functions. Specifically, given a "father" function, shaped by the state constraints, and a "mother" function, designed with some optimality criterion, the merging CLF should be similar to the father close to the constraints and similar to the mother close to the origin. To successfully merge two CLFs, the control-sharing condition is crucial: the two functions must have a common control law that makes both Lyapunov derivatives simultaneously negative. Unfortunately, it is difficult to guarantee this property a-priori, i.e., while computing the two parents functions. In this paper, we propose a technique to create a constraint-shaped "father" function that has the control-sharing property with the "mother" function. To this end, we introduce a partial control-sharing, namely, the control-sharing only in the regions where the constraints are active. We show that imposing partial control-sharing is a convex optimization problem. Finally, we show how to apply the partial control-sharing for merging constraint-shaped functions and the Riccati-optimal functions, thus generating a CLF with bounded complexity that solves the constrained linear-quadratic stabilization problem with local optimality.