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Register a new userKoopman Operator Dynamical Models: Learning, Analysis and Control
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The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigms relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.
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Methods for constructing causal linear models from nonlinear dynamical systems through lifting linearization underpinned by Koopman operator and physical system modeling theory are presented. Outputs of a nonlinear control system, called observables, may be functions of state and input, $phi(x,u)$. These input-dependent observables cannot be used for lifting the system because the state equations in the augmented space contain the time derivatives of input and are therefore anticausal. Here, the mechanism of creating anticausal observables is examined, and two methods for solving the causality problem in lifting linearization are presented. The first method is to replace anticausal observables by their integral variables $phi^*$, and lift the dynamics with $phi^*$, so that the time derivative of $phi^*$ does not include the time derivative of input. The other method is to alter the original physical model by adding a small inertial element, or a small capacitive element, so that the systems causal relationship changes. These augmented dynamics alter the signal path from the input to the anticausal observable so that the observables are not dependent on inputs. Numerical simulations validate the effectiveness of the methods.
In many applications, and in systems/synthetic biology, in particular, it is desirable to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point), or in other words to solve the switching problem. It was recently shown that, for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that this problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows to build closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system. One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. Besides the presence of numerical solutions, the switching/convergence problem can also serve as a building block for solving more complicated control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. Potentially, our approach can be extended to problems with different parametrizations of control signals since the only fundamental limitation is the finite time application of the control signal.
In this paper we prove new connections between two frameworks for analysis and control of nonlinear systems: the Koopman operator framework and contraction analysis. Each method, in different ways, provides exact and global analyses of nonlinear systems by way of linear systems theory. The main results of this paper show equivalence between contraction and Koopman approaches for a wide class of stability analysis and control design problems. In particular: stability or stablizability in the Koopman framework implies the existence of a contraction metric (resp. control contraction metric) for the nonlinear system. Further in certain cases the converse holds: contraction implies the existence of a set of observables with which stability can be verified via the Koopman framework. We provide results for the cases of autonomous and time-varying systems, as well as orbital stability of limit cycles. Furthermore, the converse claims are based on a novel relation between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger observer. We also provide a byproduct of the main results, that is, a new method to learn contraction metrics from trajectory data via linear system identification.
A graph theoretic framework recently has been proposed to stabilize interconnected multiagent systems in a distributed fashion, while systematically capturing the architectural aspect of cyber-physical systems with separate agent or physical layer and control or cyber layer. Based on that development, in addition to the modeling uncertainties over the agent layer, we consider a scenario where the control layer is subject to the denial of service attacks. We propose a step-by-step procedure to design a control layer that, in the presence of the aforementioned abnormalities, guarantees a level of robustness and resiliency for the final two-layer interconnected multiagent system. The incorporation of an event-triggered strategy further ensures an effective use of the limited energy and communication resources over the control layer. We theoretically prove the resilient, robust, and Zeno-free convergence of all state trajectories to the origin and, via a simulation study, discuss the feasibility of the proposed ideas.
We present a method for incremental modeling and time-varying control of unknown nonlinear systems. The method combines elements of evolving intelligence, granular machine learning, and multi-variable control. We propose a State-Space Fuzzy-set-Based evolving Modeling (SS-FBeM) approach. The resulting fuzzy model is structurally and parametrically developed from a data stream with focus on memory and data coverage. The fuzzy controller also evolves, based on the data instances and fuzzy model parameters. Its local gains are redesigned in real-time -- whenever the corresponding local fuzzy models change -- from the solution of a linear matrix inequality problem derived from a fuzzy Lyapunov function and bounded input conditions. We have shown one-step prediction and asymptotic stabilization of the Henon chaos.
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