We design and test a cone finding algorithm to robustly address nonlinear system analysis through... more 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. D. Kousoulidis is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. D. Kousoulidis and F. Forni are with the Department of Engineering,
The paper studies homeostatic ion channel trafficking in neurons. We derive a nonlinear closed-lo... more 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.
The concept of passivity is central to analyze circuits as interconnections of passive components... more 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.
We use the recently introduced concept of a Scaled Relative Graph (SRG) to develop a graphical an... more 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. The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.
The notion of path-complete p-dominance for switching linear systems (in short, path-dominance) i... more 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 lowdimensional dominant behavior, and hence allow for a simplified analysis of their dynamics. An algorithm for deciding the pathdominance of a given system is presented.
We discuss the role of monotonicity in enabling numerically tractable modular control design for ... more 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.
The paper studies differentially positive systems, that is, systems whose linearization along an ... more The paper studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. Extending the results in [7], 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.
The notion of path-complete positivity is introduced as a way to generalize the property of posit... more 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.
The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram that may be plotted fo... more 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.
Biomolecular feedback systems are now a central application area of interest within control theor... more 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.
Following the seminal work of Zames, the input-output theory of the 70s acknowledged that increme... more 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.
Differential positivity and K-cooperativity, a special case of differential positivity, extend di... more 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 Kcooperativity 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.
We study the problem of controlling oscillations in closed loop by combining positive and negativ... more We study the problem of controlling oscillations in closed loop by combining positive and negative feedback in a mixed configuration. We develop a complete design procedure to set the relative strength of the two feedback loops to achieve steady oscillations. The proposed design takes advantage of dominance theory and adopts classical harmonic balance and fast/slow analysis to regulate the frequency of oscillations. The design is illustrated on a simple two-mass system, a setting that reveals the potential of the approach for locomotion, mimicking approaches based on central pattern generators.
2022 IEEE 61st Conference on Decision and Control (CDC), Dec 6, 2022
We propose a new analog feedback controller based on the classical cross coupled electronic oscil... more 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.
Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they ar... more 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. D. Kousoulidis is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. D. Kousoulidis and F. Forni are with the Department of Engineering,
High-dimensional systems that have a lowdimensional dominant behavior allow for model reduction a... more High-dimensional systems that have a lowdimensional 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 particular 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.
We present a design framework that combines positive and negative feedback for robust stable osci... more We present a design framework that combines positive and negative feedback for robust stable oscillations in closed loop. The design is initially based on graphical methods, to guide the selection of the overall strength of the feedback (gain) and of the relative proportion of positive and negative feedback (balance). The design is then generalized via linear matrix inequalities. The goal is to guarantee robust oscillations to bounded dynamic uncertainties and to extend the approach to passive interconnections. The results of the paper provide a first system-theoretic justification to several observations from system biology and neuroscience pointing at mixed feedback as a fundamental enabler for robust oscillations.
We design and test a cone finding algorithm to robustly address nonlinear system analysis through... more 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. D. Kousoulidis is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. D. Kousoulidis and F. Forni are with the Department of Engineering,
The paper studies homeostatic ion channel trafficking in neurons. We derive a nonlinear closed-lo... more 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.
The concept of passivity is central to analyze circuits as interconnections of passive components... more 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.
We use the recently introduced concept of a Scaled Relative Graph (SRG) to develop a graphical an... more 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. The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.
The notion of path-complete p-dominance for switching linear systems (in short, path-dominance) i... more 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 lowdimensional dominant behavior, and hence allow for a simplified analysis of their dynamics. An algorithm for deciding the pathdominance of a given system is presented.
We discuss the role of monotonicity in enabling numerically tractable modular control design for ... more 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.
The paper studies differentially positive systems, that is, systems whose linearization along an ... more The paper studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. Extending the results in [7], 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.
The notion of path-complete positivity is introduced as a way to generalize the property of posit... more 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.
The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram that may be plotted fo... more 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.
Biomolecular feedback systems are now a central application area of interest within control theor... more 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.
Following the seminal work of Zames, the input-output theory of the 70s acknowledged that increme... more 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.
Differential positivity and K-cooperativity, a special case of differential positivity, extend di... more 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 Kcooperativity 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.
We study the problem of controlling oscillations in closed loop by combining positive and negativ... more We study the problem of controlling oscillations in closed loop by combining positive and negative feedback in a mixed configuration. We develop a complete design procedure to set the relative strength of the two feedback loops to achieve steady oscillations. The proposed design takes advantage of dominance theory and adopts classical harmonic balance and fast/slow analysis to regulate the frequency of oscillations. The design is illustrated on a simple two-mass system, a setting that reveals the potential of the approach for locomotion, mimicking approaches based on central pattern generators.
2022 IEEE 61st Conference on Decision and Control (CDC), Dec 6, 2022
We propose a new analog feedback controller based on the classical cross coupled electronic oscil... more 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.
Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they ar... more 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. D. Kousoulidis is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. D. Kousoulidis and F. Forni are with the Department of Engineering,
High-dimensional systems that have a lowdimensional dominant behavior allow for model reduction a... more High-dimensional systems that have a lowdimensional 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 particular 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.
We present a design framework that combines positive and negative feedback for robust stable osci... more We present a design framework that combines positive and negative feedback for robust stable oscillations in closed loop. The design is initially based on graphical methods, to guide the selection of the overall strength of the feedback (gain) and of the relative proportion of positive and negative feedback (balance). The design is then generalized via linear matrix inequalities. The goal is to guarantee robust oscillations to bounded dynamic uncertainties and to extend the approach to passive interconnections. The results of the paper provide a first system-theoretic justification to several observations from system biology and neuroscience pointing at mixed feedback as a fundamental enabler for robust oscillations.
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Papers by Fulvio Forni