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Systems that show different characteristics, such as finite-gain and passivity, depending on the nature of the inputs, are said to possess mixed input-output properties. In this paper, we provide a constructive method for characterizing mixed input-output properties of nonlinear systems using a dissipativity framework. Our results take inspiration from the generalized Kalman-Yakubovich-Popov lemma, and show that a system is mixed if it is dissipative with respect to highly specialized supply rates. The mixed input-output characterization is used for assessing stability of feedback interconnections in which the feedback components violate conditions of classical results such as the small-gain and passivity theorem. We showcase applicability of our results through various examples., Comment: 6 pages

Splitting algorithms are well-established in convex optimization and are designed to solve large-scale problems. Using such algorithms to simulate the behavior of nonlinear circuit networks provides scalable methods for the simulation and design of neuromorphic systems. For circuits made of linear capacitors and inductors with nonlinear resistive elements, we propose a splitting that breaks the network into its LTI lossless component and its static resistive component. This splitting has both physical and algorithmic advantages and allows for separate calculations in the time domain and in the frequency domain. To demonstrate the scalability of this approach, a network made from one hundred neurons modeled by the well-known FitzHugh-Nagumo circuit with all-to-all diffusive coupling is simulated., Comment: Submitted to CDC 2024

The paper introduces a kernel-based framework to model and identify time-invariant systems with the fading memory property. The key departure from the previous literature is to bypass the state-space representation of the model. Instead, a kernel representation is used to directly model the memory functional that maps past inputs to the present output. We explore the versatility of this approach to encode important system properties in the hyperparameters of the kernel. The approach is illustrated on the Hodgkin and Huxley model of neuronal excitability.

It is shown that an LTI system is a relaxation system if and only if its Hankel operator is cyclic monotone. Cyclic monotonicity of the Hankel operator implies the existence of a storage function whose gradient is the Hankel operator. This storage is a function of past inputs alone, is independent of the state space realization, and admits a generalization to nonlinear circuit elements., Comment: Accepted to the 2023 IEEE Conference on Decision and Control

It is shown that the behavior of an $m$-port circuit of maximal monotone elements can be expressed as a zero of the sum of a maximal monotone operator containing the circuit elements, and a structured skew-symmetric linear operator representing the interconnection structure, together with a linear output transformation. The Condat-V\~u algorithm solves inclusion problems of this form, and may be used to solve for the periodic steady-state behavior, given a periodic excitation at each port, using an iteration in the space of periodic trajectories., Comment: Published in the European Journal of Control special issue for the 2023 European Control Conference

The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram that may be plotted for nonlinear operators, and allows nonlinear robustness margins to be defined graphically. This abstract explores techniques for shaping the SRG of an operator in order to maximize these robustness margins., Comment: To be presented at MTNS 2022

Following the seminal work of Zames, the input-output theory of the 70s acknowledged that incremental properties (e.g. incremental gain) are the relevant quantities to study in nonlinear feedback system analysis. Yet, non-incremental analysis has dominated the use of dissipativity theory in nonlinear control from the 80s. Results connecting dissipativity theory and incremental analysis are scattered and progress has been limited. This abstract investigates whether this theoretical gap is of fundamental nature and considers new avenues to circumvent it., Comment: To be presented at MTNS 2022

The splitting algorithms of monotone operator theory find zeros of sums of relations. This corresponds to solving series or parallel one-port electrical circuits, or the negative feedback interconnection of two subsystems. One-port circuits with series and parallel interconnections, or block diagrams with multiple forward and return paths, give rise to current-voltage relations consisting of nested sums and inverses. In this extended abstract, we present new splitting algorithms specially suited to these structures, for interconnections of monotone and anti-monotone relations., Comment: To be presented at MTNS 2022

Continued fractions are classical representations of complex objects (for example, real numbers) as sums and inverses of simpler objects (for example, integers). The analogy in linear circuit theory is a chain of series/parallel one-ports: the port behavior is a continued fraction containing the port behaviors of its elements. Truncating a continued fraction is a classical method of approximation, which corresponds to deleting the circuit elements furthest from the port. We apply this idea to chains of series/parallel one-ports composed of arbitrary nonlinear relations. This gives a model reduction method which automatically preserves properties such as incremental positivity. The Scaled Relative Graph (SRG) gives a graphical representation of the original and truncated port behaviors. The difference of these SRGs gives a bound on the approximation error, which is shown to be competitive with existing methods., Comment: Submitted to CDC2022

We propose a new analog feedback controller based on the classical cross coupled electronic oscillator. The goal is to drive a linear passive plant into oscillations. We model the circuit as Lur'e system and we derive a new graphical condition to certify oscillations (inverse circle criterion for dominance theory). These conditions are then specialized to minimal control architectures like RLC and RC networks, and are illustrated with an example based on a DC motor model.

Maximal monotonicity is explored as a generalization of the linear theory of passivity, aiming at an algorithmic input/output analysis of physical models. The theory is developed for maximal monotone one-port circuits, formed by the series and parallel interconnection of basic elements. These circuits generalize passive LTI transfer functions. Periodic input signals are shown to be mapped to periodic output signals, and these input-output behaviors can be efficiently computed using a maximal monotone splitting algorithm, which decomposes the computation according to the circuit topology. A new splitting algorithm is presented, which applies to any monotone one-port circuit defined as a port interconnection of monotone elements.

Given two nonlinear systems which only violate incremental passivity when their incremental gains are sufficiently small, we give a condition for their negative feedback interconnection to have finite incremental gain, which generalizes the incremental small gain and incremental passivity theorems. The property may be determined graphically by plotting the Scaled Relative Graphs (SRGs) of the systems, which provides engineering significance to the mathematical result.

We use the recently introduced concept of a Scaled Relative Graph (SRG) to develop a graphical analysis of input-output properties of feedback systems. The SRG of a nonlinear operator generalizes the Nyquist diagram of an LTI system. In the spirit of classical control theory, important robustness indicators of nonlinear feedback systems are measured as distances between SRGs., Comment: This work has been submitted to the IEEE for possible publication

A new method is presented for the analysis of limit cycle oscillations in mixed-feedback systems. The calculation of the limit cycle is reformulated as the zero finding of a mixed-monotone relation, that is, of the difference of two maximally monotone relations. The problem can then be solved efficiently by borrowing existing algorithms that minimize the difference of two convex functions. The potential of the method is illustrated on the classical Van der Pol oscillator.

Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on $L_2$ is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem., Comment: Accepted to the 2021 IEEE Conference on Decision and Control

The circuit-theoretic origins of maximal monotonicity are revisited using modern optimization algorithms for maximal monotone operators. We present an algorithm for computing the periodic behavior of an interconnection of maximal monotone systems using a fixed point iteration. The fixed point iteration may be split according to the interconnection structure of the system. In this preliminary work, the approach is demonstrated on port interconnections of maximal monotone resistors and LTI capacitors and inductors., Comment: Accepted for presentation at the 2021 European Control Conference

Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provide open loop and sampled data controllers. The sampled data control construction presented here does not require real time computation of globally shortest paths, simplifying computation., Comment: accepted to 2018 American Control Conference

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show integrability of differential control signals. This technical condition is shown to be satisfied under the original assumptions if the input matrix is constant rank, and also if the strong conditions for a CCM hold. However a simple counterexample shows that if the input matrix drops rank, then the weaker conditions of the original theorem may not imply stabilizability of all trajectories. The remaining claims and illustrative examples of the paper are shown to remain valid with the new condition., Comment: Amendment to: I. R. Manchester, J. J. E. Slotine Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design, IEEE Transactions on Automatic Control, Volume 62, Issue 6, pp 3046-3053 June 2017