Search

We illustrate the potential of neuromorphic control on the simple mechanical model of a pendulum, with both event-based actuation and sensing. The controller and the pendulum are regarded as event-based systems that occasionally interact to coordinate their respective rhythms. Control occurs through a proper timing of the interacting events. We illustrate the mixed nature of the control design: the design of a rhythmic automaton, able to generate the right sequence of events, and the design of a feedback regulator, able to tune the timing of events., Comment: 6 pages, 10 figures. Accepted by IEEE Control Systems Letters

We consider the design of a new class of passive iFIR controllers given by the parallel action of an integrator and a finite impulse response filter. iFIRs are more expressive than PID controllers but retain their features and simplicity. The paper provides a model-free data-driven design for passive iFIR controllers based on virtual reference feedback tuning. Passivity is enforced through constrained optimization (three different formulations are discussed). The proposed design does not rely on large datasets or accurate plant models., Comment: 6 pages, 7 figures, Accepted by IEEE Control Systems Letters (L-CSS) with the option to present it to 2024 Conference on Decision and Control (CDC 2024)

Energy based control methods are at the core of modern robotic control algorithms. In this paper we present a general approach to virtual model/mechanism control, which is a powerful design tool to create energy based controllers. We present two novel virtual-mechanisms designed for robotic minimally invasive surgery, which control the position of a surgical instrument while passing through an incision. To these virtual mechanisms we apply the parameter tuning method of Larby and Forni 2022, which optimizes for local performance while ensuring global stability., Comment: 8 pages, 10 figures. Submitted to IFAC 2023; The 22nd World Congress of the International Federation of Automatic Control

Most impedance control schemes in robotics implement a desired passive impedance, allowing for stable interaction between the controlled robot and the environment. However, there is little guidance on the selection of the desired impedance. In general, finding the best stiffness and damping parameters is a challenging task. This paper contributes to this problem by connecting impedance control to robust control, with the goal of shaping the robot performances via feedback. We provide a method based on linear matrix inequalities with sparsity constraints to derive impedance controllers that satisfy a H-infinity performance criterion. Our controller guarantees passivity of the controlled robot and local performances near key poses., Comment: 7 pages, 6 figures. To be published in proceedings of the IEEE Conference on Decision and Control (CDC) 2022

Improving the efficiency of agriculture is a growing priority due to food security issues, environmental concerns, and economics. Precision agriculture and variable rate application technology could enable increases in yield while maintaining or reducing fertiliser use. However, this requires the development of control algorithms which are suitable for the challenges of agriculture. In this paper, we propose a new mechanistic open model of lettuce growth for use in control of precision agriculture. We demonstrate that our model is cooperative and fits well to experimental data. We use the model to show, via simulations, that a simple proportional distributed control law increases crop uniformity and yield without increasing nitrogen use, even in the presence of sparse actuation and noisy observations., Comment: 8 pages, Submitted to ECC23

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

We discuss the role of monotonicity in enabling numerically tractable modular control design for networked nonlinear systems. We first show that the variational systems of monotone systems can be embedded into positive systems. Utilizing this embedding, we show how to solve a network stabilization problem by enforcing monotonicity and exponential dissipativity of the network sub-components. Such modular approach leads to a design algorithm based on a sequence of linear programming problems.

We provide novel dissipativity conditions for bounding the incremental L-1 gain of systems. Moreover, we adapt existing results on the L-infinity gain to the incremental setting and relate the incremental L-1 and L-infinity gain bounds through system adjoints. Building on work on optimization based approaches to constructing polyhedral Lyapunov functions, we make use of these conditions to obtain a Linear Programming based algorithm that can provide increasingly sharp bounds on the gains as a function of a given candidate polyhedral storage function or polyhedral set. The algorithm is also extended to allow for the design of linear feedback controllers for performance, as measured by the bounds on the incremental gains. We apply the algorithm to a couple of numerical examples to illustrate the power, as well as some limitations, of this approach.

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.

We present a design framework to induce stable oscillations through mixed feedback control. We provide conditions on the feedback gain and on the balance between positive and negative feedback contributions to guarantee robust oscillations. Using linear matrix inequalities, we later derive a systematic design for robustness to bounded dynamic uncertainties and for passive interconnections. The results of the paper provide a system-theoretic justification to several observations from system biology and neuroscience pointing at the mixed feedback as a fundamental enabler for robust oscillations. Our results are illustrated through a distributed electrical circuit mimicking (simplified) neural dynamics.

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. Copyright may be transferred without notice, after which this version may no longer be accessible

Neurons regulate the distribution of signaling components across an extended tree-like cellular structure using both local and global feedback control. This is hypothesized to allow homeostatic control of the electrical activity of a neuron and at the same time enable normalization of distribution of inputs received from other cells. The performance and robustness of these mechanisms are poorly understood, and are subject to nonlinearities, making their analysis difficult. Firstly, we formally show that global homeostasis of electrical activity and local activity-dependent degradation can coexist under sufficient timescale separation. The interplay of the two feedback mechanisms is also analyzed through simulations, which reveal a bidirectional effect (stabilizing and destabilizing) of activity-dependent degradation on the overall neuron performance., Comment: 6 pages; accepted to the 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2020)

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

Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they are used to study the stability of linear time varying and linear parameter varying systems without being conservative. However, the computational cost associated with using them grows unbounded as the size of their representation increases. Finding them is also a hard computational problem. Here we present an algorithm that attempts to find polyhedral functions while keeping the size of the representation fixed, to limit computational costs. We do this by measuring the gap from contraction for a given polyhedral set. The solution is then used to find perturbations on the polyhedral set that reduce the contraction gap. The process is repeated until a valid polyhedral Lyapunov function is obtained. The approach is rooted in linear programming. This leads to a flexible method capable of handling additional linear constraints and objectives, and enables the use of the algorithm for control synthesis.

Neuronal circuits internally regulate electrical signaling via a host of homeostatic mechanisms. Two prominent mechanisms, synaptic scaling and structural plasticity, are believed to maintain average activity within an operating range by modifying the strength and spatial extent of network connectivity using negative feedback. However, both mechanisms operate on relatively slow timescales and thus face fundamental limits due to delays. We show that these mechanisms fulfill complementary roles in maintaining stability in a large network. In particular, even relatively, slow growth dynamics improves performance significantly beyond synaptic scaling alone., Comment: Accepted to the European Control Conference 2021

The interplay of positive and negative feedback loops on different time scales appears to be a fundamental mechanisms for robust and tunable oscillations in both biological systems and electro-mechanical systems. We develop a detailed analysis of a basic three dimensional Lure model to show how controlled oscillations arise from the tuning of positive and negative feedback strengths. Our analysis is based on dominance theory and confirms, from a system-theoretic perspective, that the mixed feedback is a fundamental enabler of robust oscillations. Our results are not limited to three dimensional systems and extend to larger systems via passivity theory, and to uncertain systems via small gain arguments., Comment: Submitted to European Control Conference 2021

The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, model reduction methods are not well developed for non-equilibrium behaviors. The proposed framework addresses this need by exploiting recent advances on dominance theory. Classical balanced truncation for linear time-invariant systems is used to develop a dominance-preserving model reduction method for Lure systems, i.e. systems that can be decomposed as the feedback interconnection of a linear system and a static nonlinearity. The method is illustrated by approximating the oscillatory behavior of a discretized heat flow control system., Comment: 6 pages, 6 figures, submitted to the IFAC World Congress 2020

Systems whose variable are constrained to be positive allow computationally efficient control design. We generalize these results to linear systems which leave a cone invariant. This is a wider class of systems than positive systems. We revisit classical results on stability and dissipativity of positive linear systems and show how scalable conditions on linear programming can be extended to cone invariant linear systems. Our results are illustrated by scalable stabilizing controller design for mass-spring systems.

Differential positivity and K-cooperativity, a special case of differential positivity, extend differential approaches to control to nonlinear systems with multiple equilibria, such as switches or multi-agent consensus. To apply this theory, we reframe conditions for strict K-cooperativity as an optimization problem. Geometrically, the conditions correspond to finding a cone that a set of linear operators leave invariant. Even though solving the optimization problem is hard, we combine the optimization perspective with the geometric intuition to construct a heuristic cone-finding algorithm centered around Linear Programming (LP). The algorithm we obtain is unique in that it modifies existing rays of a candidate cone instead of adding new ones. This enables us to also take a first step in tackling the synthesis problem for K-cooperative systems. We demonstrate our approach on some examples, including one in which we repurpose our algorithm to obtain a novel alternative tool for computing polyhedral Lyapunov functions of bounded complexity., Comment: 8 pages, to be presented in the 21st IFAC World Congress

Biomolecular feedback systems are now a central application area of interest within control theory. While classical control techniques provide invaluable insight into the function and design of both natural and synthetic biomolecular systems, there are certain aspects of biological control that have proven difficult to analyze with traditional methods. To this end, we describe here how the recently developed tools of dominance analysis can be used to gain insight into the nonlinear behavior of the antithetic integral feedback circuit, a recently discovered control architecture which implements integral control of arbitrary biomolecular processes using a simple feedback mechanism. We show that dominance theory can predict both monostability and periodic oscillations in the circuit, depending on the corresponding parameters and architecture. We then use the theory to characterize the robustness of the asymptotic behavior of the circuit in a nonlinear setting., Comment: 7 pages, 7 figures

The differential $\mathcal{L}_{2,p}$ gain of a linear, time-invariant, $p$-dominant system is shown to coincide with the $\mathcal{H}_{\infty,p}$ norm of its transfer function $G$, defined as the essential supremum of the absolute value of $G$ over a vertical strip in the complex plane such that $p$ poles of $G$ lie to right of the strip. The close analogy between the $\mathcal{H}_{\infty,p}$ norm and the classical $\mathcal{H}_{\infty}$ norm suggests that robust dominance of linear systems can be studied along the same lines as robust stability. This property can be exploited in the analysis and design of nonlinear uncertain systems that can be decomposed as the feedback interconnection of a linear, time-invariant system with bounded gain uncertainties or nonlinearities., Comment: 6 pages, 3 figures, 58th IEEE Conf. Decision and Control

The paper studies homeostatic ion channel trafficking in neurons. We derive a nonlinear closed-loop model that captures active transport with degradation, channel insertion, average membrane potential activity, and integral control. We study the model via dominance theory and differential dissipativity to show when steady regulation gives way to pathological oscillations. We provide quantitative results on the robustness of the closed loop behavior to static and dynamic uncertainties, which allows us to understand how cell growth interacts with ion channel regulation., Comment: 6 pages, 58th IEEE Conference on Decision and Control

This paper deals with the analysis of nonlinear circuits that interconnect passive elements (capacitors, inductors, and resistors) with nonlinear resistors exhibiting a range of $\it{negative}$ resistance. Such active elements are necessary to design circuits that switch and oscillate. We generalize the classical passivity theory of circuit analysis to account for such non-equilibrium behaviors. The approach closely mimics the classical methodology of (incremental) dissipativity theory, but with dissipation inequalities that combine $\it{signed}$ storage functions and $\it{signed}$ supply rates to account for the mixture of passive and active elements.

The paper introduces notions of robustness margins geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and in engineering, a theory of robustness for behaviors away from equilibria is lacking. The proposed framework addresses this need in the framework of p-dominance theory, which aims at generalizing stability theory for the analysis of systems with low-dimensional attractors. Dominance margins are introduced as natural generalisations of stability margins in the context of p-dominance analysis. In analogy with stability margins, dominance margins are shown to admit simple interpretations in terms of familiar frequency domain tools and to provide quantitative measures of robustness for multistable and oscillatory behaviors in Lure systems. The theory is illustrated by means of an elementary mechanical example., Comment: 11th IFAC Symposium on Nonlinear Control Systems

We design and test a cone finding algorithm to robustly address nonlinear system analysis through differential positivity. The approach provides a numerical tool to study multi-stable systems, beyond Lyapunov analysis. The theory is illustrated on two examples: a consensus problem with some repulsive interactions and second order agent dynamics, and a controlled duffing oscillator., Comment: 6 pages, to be presented in the 58th IEEE Conference on Decision and Control (CDC)

The notion of path-complete $p$-dominance for switching linear systems (in short, path-dominance) is introduced as a way to generalize the notion of dominant/slow modes for LTI systems. Path-dominance is characterized by the contraction property of a set of quadratic cones in the state space. We show that path-dominant systems have a low-dimensional dominant behavior, and hence allow for a simplified analysis of their dynamics. An algorithm for deciding the path-dominance of a given system is presented., Comment: 6 pages, 5 figures, to be presented at IEEE Conference on Decision and Control 2018

The concept of passivity is central to analyze circuits as interconnections of passive components. We illustrate that when used differentially, the same concept leads to an interconnection theory for electrical circuits that switch and oscillate as interconnections of passive components with operational amplifiers (op-amps). The approach builds on recent results on dominance and p-passivity aimed at generalizing dissipativity theory to the analysis of non-equilibrium nonlinear systems. Our paper shows how those results apply to basic and well-known nonlinear circuit architectures. They illustrate the potential of dissipativity theory to design and analyze switching and oscillating circuits quantitatively, very much like their linear counterparts., Comment: 6 pages, 10 figures. Submitted to Second Conference on Modelling,Identification and Control of Nonlinear Systems, IFAC MICNON 2018

The paper extends the concepts of dominance and p-dissipativity to the non-smooth family of linear complementarity systems. Dominance generalizes incremental stability whereas p-dissipativity generalizes incremental passivity. The generalization aims at an interconnection theory for the design and analysis of switching and oscillatory systems. The approach is illustrated by a detailed study of classical electrical circuits that switch and oscillate., Comment: 7 pages, 8 figures, Submitted to the 23rd MTNS Conference

High-dimensional systems that have a low-dimensional dominant behavior allow for model reduction and simplified analysis. We use differential analysis to formalize this important concept in a nonlinear setting. We show that dominance can be studied through linear dissipation inequalities and an interconnection theory that closely mimics the classical analysis of stability by means of dissipativity theory. In this approach, stability is seen as the limiting situation where the dominant behavior is 0-dimensional. The generalization opens novel tractable avenues to study multistability through 1-dominance and limit cycle oscillations through 2-dominance., Comment: 11 pages, 7 figures, IEEE Transaction of Automatic control

We revisit the classical dissipativity theorem of linear-quadratic theory in a generalized framework where the quadratic storage is negative definite in a p-dimensional subspace and positive definite in a complementary subspace. The classical theory assumes p = 0 and provides an inter- connection theory for stability analysis, i.e. convergence to a zero dimensional attractor. The generalized theory is shown to provide an interconnection theory for p-dominance analysis, i.e. convergence to a p-dimensional dominant subspace. In turn, this property is the differential characterization of a generalized contraction property for nonlinear systems. The proposed generalization opens a novel avenue for the analysis of interconnected systems with low-dimensional attractors., Comment: 6 pages, 2 figures, in 56th IEEE Conference on Decision and Control (CDC 2017)

The notion of path-complete positivity is introduced as a way to generalize the property of positivity from one LTI system to a family of switched LTI systems whose switching rule is constrained by a finite automaton. The generalization builds upon the analogy between stability and positivity, the former referring to the contraction of a norm, the latter referring to the contraction of a cone (or, equivalently, a projective norm). We motivate and investigate the potential of path-positivity and we propose an algorithm for the automatic verification of positivity., Comment: 7 pages

The paper shows that normally hyperbolic one-dimensional compact attractors of smooth dynamical systems are characterized by differential positivity, that is, the pointwise infinitesimal contraction of a smooth cone field. The result is analog to the characterization of zero-dimensional hyperbolic attractors by differential stability, which is the pointwise infinitesimal contraction of a Riemannian metric.

We study the existence of asymptotically stable periodic trajectories induced by reset feedback. The analysis is developed for a planar system. Casting the problem into the hybrid setting, we show that a periodic orbit arises from the balance between the energy dissipated during flows and the energy restored by resets, at jumps. The stability of the periodic orbit is studied with hybrid Lyapunov tools. The satisfaction of the so-called hybrid basic conditions ensures the robustness of the asymptotic stability. Extensions of the approach to more general mechanical systems are discussed., Comment: 7 pages, 16 figures

The paper studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. We illustrate the use of differential positivity on compact forward invariant sets for the characterization of bistable and periodic behaviors. Geometric conditions for differential positivity are provided. The introduction of compact sets simplifies the use of differential positivity in applications., Comment: 6 pages, 3 figures

The main objective of this paper is to propose an alternative procedure to carry out one of the key steps of immersion and invariance stabilising controller design. Namely, the one that ensures attractivity of the manifold whose internal dynamics contains a copy of the desired system behaviour. Towards this end we invoke contraction theory principles and ensure the attractivity of the manifold rendering it horizontally contractive. The main advantage of adopting this alternative approach is to make more systematic the last step of the design with more explicit degrees of freedom to accomplish the task. The classical case of systems in feedback form is used to illustrate the proposed controller design., Comment: 5 pages, the 54th IEEE Conference on Decision and Control

Differential analysis aims at inferring global properties of nonlinear behaviors from the local analysis of the linearized dynamics. The paper motivates and illustrates the use of differential analysis on the nonlinear pendulum model, an archetype example of nonlinear behavior. Special emphasis is put on recent work by the authors in this area, which includes a differential Lyapunov framework for contraction analysis and the concept of differential positivity., Comment: 12 pages, 14 figures. Tutorial paper for the 53rd IEEE Conference on Decision and Control, 2014

The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron Frobenius theory is developed in this differential framework to show that the property induces a (conal) order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space., Comment: 32 pages, 10 figures

We consider an exponentially stable closed loop interconnection of a continuous linear plant and a continuous linear controller, and we study the problem of interconnecting the plant output to the controller input through a digital channel. We propose a family of "transmission-lazy" sensors whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. In particular, we propose two transmission policies, providing conditions on the transmission parameters. These guarantee global asymptotic stability when the plant state is available or when an estimate of the state is available (provided by a classical continuous linear observer). Moreover, under a specific condition, they guarantee global exponential stability, Comment: 16 pages

Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of regulation, synchronization, and estimation. The paper investigates how restrictive is the property, focusing on a class of open gradient systems encountered in the coenergy modeling framework of physical systems, in particular the Brayton-Moser formalism for nonlinear electrical circuits., Comment: 52nd IEEE Conference on Decision and Control, 2013

Dissipativity is an essential concept of systems theory. The paper provides an extension of dissipativity, named differential dissipativity, by lifting storage functions and supply rates to the tangent bundle. Differential dissipativity is connected to incremental stability in the same way as dissipativity is connected to stability. It leads to a natural formulation of differential passivity when restricting to quadratic supply rates. The paper also shows that the interconnection of differentially passive systems is differentially passive, and provides preliminary examples of differentially passive electrical systems.

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves., Comment: 15 pages, 1 figure