Switching Dynamical Systems Research Articles

Second-Order Partial Outer Convexification for Switched Dynamical Systems

Second-Order Partial Outer Convexification for Switched Dynamical Systems

Optimal Control of Switched Dynamical Systems Under Dwell Time Constraints - Theory and Computation

Optimal Control of Switched Dynamical Systems Under Dwell Time Constraints - Theory and Computation

Unsupervised learning of stationary and switching dynamical system models from Poisson observations

Objective. Investigating neural population dynamics underlying behavior requires learning accurate models of the recorded spiking activity, which can be modeled with a Poisson observation distribution. Switching dynamical system models can offer both explanatory power and interpretability by piecing together successive regimes of simpler dynamics to capture more complex ones. However, in many cases, reliable regime labels are not available, thus demanding accurate unsupervised learning methods for Poisson observations. Existing learning methods, however, rely on inference of latent states in neural activity using the Laplace approximation, which may not capture the broader properties of densities and may lead to inaccurate learning. Thus, there is a need for new inference methods that can enable accurate model learning. Approach. To achieve accurate model learning, we derive a novel inference method based on deterministic sampling for Poisson observations called the Poisson Cubature Filter (PCF) and embed it in an unsupervised learning framework. This method takes a minimum mean squared error approach to estimation. Terms that are difficult to find analytically for Poisson observations are approximated in a novel way with deterministic sampling based on numerical integration and cubature rules. Main results. PCF enabled accurate unsupervised learning in both stationary and switching dynamical systems and largely outperformed prior Laplace approximation-based learning methods in both simulations and motor cortical spiking data recorded during a reaching task. These improvements were larger for smaller data sizes, showing that PCF-based learning was more data efficient and enabled more reliable regime identification. In experimental data and unsupervised with respect to behavior, PCF-based learning uncovered interpretable behavior-relevant regimes unlike prior learning methods. Significance. The developed unsupervised learning methods for switching dynamical systems can accurately uncover latent regimes and states in population spiking activity, with important applications in both basic neuroscience and neurotechnology.

Generalized sliding event‐triggered control for dissipative performance of Markov switching dynamic systems

SummaryIn this paper, a generalized sliding event‐triggered control for the closed‐loop Markov switching systems (CLMSSs) is studied. It is proved that the system is driven to the designed sliding mode surface under a given sliding mode controller. A generalized sliding event‐triggered controller is developed to obtain strictly dissipative performance of CLMSSs, in which the combination of sliding mode control and event‐triggering protocol makes the control performance more efficient and generalized. Further, two dissipative criteria for CLMSSs are established. Finally, the results of numerical simulations give a witness for the effectiveness of the given theorems and methods.

Development of brain state dynamics involved in working memory.

Human functional brain networks are dynamically organized to enable cognitive and behavioral flexibility to meet ever-changing environmental demands. Frontal-parietal network (FPN) and default mode network (DMN) are recognized to play an essential role in executive functions such as working memory. However, little is known about the developmental differences in the brain-state dynamics of these two networks involved in working memory from childhood to adulthood. Here, we implemented Bayesian switching dynamical systems approach to identify brain states of the FPN and DMN during working memory in 69 school-age children and 51 adults. We identified five brain states with rapid transitions, which are characterized by dynamic configurations among FPN and DMN nodes with active and inactive engagement in different task demands. Compared with adults, children exhibited less frequent brain states with the highest activity in FPN nodes dominant to high demand, and its occupancy rate increased with age. Children preferred to attain inactive brain states with low activity in both FPN and DMN nodes. Moreover, children exhibited lower transition probability from low-to-high demand states and such a transition was positively correlated with working memory performance. Notably, higher transition probability from low-to-high demand states was associated with a stronger structural connectivity across FPN and DMN, but with weaker structure-function coupling of these two networks. These findings extend our understanding of how FPN and DMN nodes are dynamically organized into a set of transient brain states to support moment-to-moment information updating during working memory and suggest immature organization of these functional brain networks in childhood, which is constrained by the structural connectivity.

Data-Driven Iterative Optimal Control for Switched Dynamical Systems

This article presents a data-driven algorithm to compute optimal control inputs for input-constrained nonlinear optimal control problems with switched dynamics. We consider multi-stage optimal control problems where the control inputs and the switching instants are both unknown. Our key contribution is the new iterative online optimal control algorithm which mitigates sub-optimal control caused by model bias in the challenging class of under-actuated and intrinsically unstable switched dynamical systems. This is achieved by estimating the cost and computing the control inputs along measured trajectories of the controlled system instead of doing the same procedure along error-prone trajectories predicted by an inexact model. The algorithm is evaluated using an under-actuated and intrinsically unstable hopping robot in a simulation environment. The algorithm enables real-time data-driven optimal control using inaccurate models.

A Switched Dynamical System Approach to the Flow and Heat Transfer of Immiscible Fluids

Abstract This paper provides important insights into immiscible fluids models, which can exhibit complex nonlinear behavior due to variations in thickness, viscosity, thermal conductivity, and jump velocity fields across the interface surface. These characteristics will be used to model our problem as a switching dynamical system. The bifurcation tools for switching systems will be used to create a systematic computational analysis of the rise dynamics of the behavior of solutions influenced by the interface surface. This technique is applied to investigate the flow and heat transfer behavior of two immiscible fluids for a recently proposed model. The explicit formula for tracking the accurate behavior of the interface surface, which is used as a critical part of the bifurcation analysis, is provided. The results show that the existence of heteroclinic connections and switching stability of multiple equilibria are the primary causes of the formation of a novel class of trapping phenomena. The biological significance of our results on flow and heat transfer characteristics is discussed.

Variational Inference and Learning of Piecewise Linear Dynamical Systems.

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because their computational complexity increases exponentially with time. In this article, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms: a filter and a smoother. We show that the model parameters can be split into two sets: static and dynamic parameters, and that the former parameters can be estimated offline together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to the head-pose tracking, and we thoroughly compare our algorithms with several state of the art trackers.

Models to assess imported cases on the rebound of COVID-19 and design a long-term border control strategy in Heilongjiang Province, China.

Since the outbreak of COVID-19 in Wuhan, China in December 2019, it has spread quickly and become a global pandemic. While the epidemic has been contained well in China due to unprecedented public health interventions, it is still raging or not yet been restrained in some neighboring countries. Chinese government adopted a strict policy of immigration diversion in major entry ports, and it makes Suifenhe port in Heilongjiang Province undertook more importing population. It is essential to understand how imported cases and other key factors of screening affect the epidemic rebound and its mitigation in Heilongjiang Province. Thus we proposed a time switching dynamical system to explore and mimic the disease transmission in three time stages considering importation and control. Cross validation of parameter estimations was carried out to improve the credibility of estimations by fitting the model with eight time series of cumulative numbers simultaneous. Simulation of the dynamics shows that illegal imported cases and imperfect protection in hospitals are the main reasons for the second epidemic wave, the actual border control intensities in the province are relatively effective in early stage. However, a long-term border closure may cause a paradox phenomenon such that it is much harder to restrain the epidemic. Hence it is essential to design an effective border reopening strategy for long-term border control by balancing the limited resources on hotel rooms for quarantine and hospital beds. Our results can be helpful for public health to design border control strategies to suppress COVID-19 transmission.

Latent brain state dynamics and cognitive flexibility in older adults.

Cognitive impairment in older adults is a rapidly growing public health concern as the elderly population dramatically grows worldwide. While it is generally assumed that cognitive deficits in older adults are associated with reduced brain flexibility, quantitative evidence has been lacking. Here, we investigate brain flexibility in healthy older adults (ages 60-85) using a novel Bayesian switching dynamical system algorithm and ultrafast temporal resolution (TR = 490 ms) whole-brain fMRI data during performance of a Sternberg working memory task. We identify latent brain states and characterize their dynamic temporal properties, including state transitions, associated with encoding, maintenance, and retrieval. Crucially, we demonstrate that brain inflexibility is associated with slower and more fragmented transitions between latent brain states, and that brain inflexibility mediates the relation between age and cognitive inflexibility. Our study provides a novel neurocomputational framework for investigating latent dynamic circuit processes underlying brain flexibility and cognition in the context of aging.

Non-smooth Modeling and Variable Structure Control of a Class of Hybrid Dynamical Systems

In this work, we propose a modeling formulation and controller design for a class of hybrid dynamical systems. In this formulation, a switching dynamical system is modeled as a dynamical system with discontinuous right hand side. More specifically, the system is transformed to a nonlinear system with discontinuous nonlinearities. Then, a synthesis of feedback linearization and sliding mode control is employed for output tracking control problem. Application and implementation of this approach is illustrated via a chemical process example.

A Method of Finding Discrete Controls for Switched Dynamical Systems by Applying Continued Fractions

Control systems with a finite number of control settings are considered. It is assumed that any polysystem operates in continuous time and control switchings occur at some certain discrete time instants. A control goal is to transfer a polysystem from an initial state to a final state. Controllability of systems switched in discrete time is studied. Controls are constructed by using the theory of generalized multicomponent continued fractions and the congruences theory. Applications of the proposed control method to specific systems are discussed.

Incentive-based decentralized routing for connected and autonomous vehicles using information propagation

Routing strategies under the aegis of dynamic traffic assignment have been proposed in the literature to optimize system performance. However, challenges have persisted in their deployment ability and effectiveness due to inherent strong assumptions on traveler behavior and availability of network-level real-time traffic information, and the high computational burden associated with computing network-wide flows in real-time. To address these gaps, this study proposes an incentive-based decentralized routing strategy to nudge the network performance closer to the system optimum for the context where all vehicles are connected and autonomous vehicles (CAVs). The strategy consists of three stages. The first stage incorporates a local route switching dynamical system to approximate the system optimal route flow in a local area based on vehicles’ knowledge of local traffic information. This system is decentralized in the sense that it only updates the local route choices of vehicles in this area rather than route choices of all vehicles in the network, which circumvents the high computational burden associated with computing the flows on the entire network. The second stage optimizes the route for each CAV by considering individual heterogeneity in traveler preferences (e.g., the value of time) to maximize the utilities of all travelers in the local area. Constraints are also incorporated to ensure that these routes can achieve the approximated local system optimal flow of the first stage. The third stage leverages an expected envy-free incentive mechanism to ensure that travelers in the local area can accept the optimal routes determined in the second stage. The study analytically discusses the convergence of the local route switching dynamical system. We also show that the proposed incentive mechanism is expected individual rational and budget-balanced, which ensure that travelers are willing to participate and guarantee the balance between payments and compensations, respectively. Further, the conditions for the expected incentive compatibility of the incentive mechanism are analyzed and proved, ensuring behavioral honesty in disclosing information. Thereby, the proposed incentive-based decentralized routing strategy can enhance network performance and user satisfaction under fully connected and autonomous environments.

Driving Mode Optimization for Hybrid Trucks Using Road and Traffic Preview Data

This paper proposes a predictive driver coaching (PDC) system for fuel economy driving for hybrid electric trucks using upcoming static map and dynamic traffic data. Unlike traditional methods that optimize over engine torque and brake to obtain a speed profile, we propose to optimize over driving modes of trucks to achieve a trade-off between fuel consumption and trip time. The optimal driving mode is provided to the driver as a coaching recommendation. To obtain the optimal solution, the truck dynamics are firstly modeled as a hybrid controlled switching dynamical system with autonomous subsystems and then a hybrid optimal control problem (HOCP) is formulated. The problem is solved using an algorithm based on discrete hybrid minimum principle. A warm-start strategy to reduce algorithmic iterations is used by employing a shrinking horizon strategy. In addition, an extensive analysis of the proposed algorithm is provided. We prove that the the coasting mode is never optimal given the truck configuration and and we provide a guideline for tuning parameters to maintain the optimal mode sequence. Finally, the algorithm is validated using real-world data from baseline driving tests using a DAF hybrid truck. Significant reduction in fuel consumption is achieved when the data is perfectly available.

Model Reference Adaptive Control of Switched Dynamical Systems with Applications to Aerial Robotics

This paper presents an adaptive control law for unknown nonlinear switched plants that must follow the trajectory of user-defined linear switched reference models. The effectiveness of the proposed control architecture is proven in two alternative frameworks, that is, analyzing Caratheodory and Filippov solutions of discontinuous differential equations. Numerical and experimental data verify the applicability of the theoretical results to problems of practical interest. The proposed numerical simulation involves the design of a model reference adaptive control law to regulate the roll dynamics of a reconfigurable delta-wing aircraft. The proposed flight tests involve an aerial robot tasked with autonomously mounting a camera of unknown inertial properties to a vertical surface.

Analysis of Switched Dynamical Systems in Perspective of Bifurcation and Multiobjective Optimization

Analysis of Switched Dynamical Systems in Perspective of Bifurcation and Multiobjective Optimization

A Bi-level Approach to MPC for Switching Nonlinear Systems

In this paper, a nonlinear model predictive control scheme for switching dynamical systems is presented. The controller comprises of two layers of optimization. The upper layer is based on the embedding transformation technique, hence it does not require prior knowledge of the switching sequence. In particular, it provides the optimal relaxed switching sequence together with the optimal regulating inputs and the corresponding trajectories of the states. Within the lower layer, the integrality constraints are restored and a switching solution is recomputed to minimize the error with respect to the trajectories given from the upper-level optimization. The scheme is presented and the bounds of the integer approximation errors are evaluated together with brief recursive feasibility analysis. Simulation results of a tracking and an economics optimizing nonlinear MPC for a supermarket refrigerator system show the applicability and efficiency of the proposed approach.

Lyapunov Characterization and Analysis of the Operating Modes of the AC–DC Ćuk Converter

In this paper, all operating modes of the ac–dc Cuk converter have been modeled as linear-time invariant systems where some of the modes were not only modeled but also detected for the first time. Another feature provided in this paper is that the converter can be modeled and analyzed as a switching dynamical system controlled by an external constant duty-ratio switching and the internal switchings shown to be related to the minimum among the time derivatives of a common quadratic Lyapunov function. Some representative simulations are provided to illustrate and validate the technical results of this paper.

Uncovering hidden brain state dynamics that regulate performance and decision-making during cognition

Human cognition is influenced not only by external task demands but also latent mental processes and brain states that change over time. Here, we use novel Bayesian switching dynamical systems algorithm to identify hidden brain states and determine that these states are only weakly aligned with external task conditions. We compute state transition probabilities and demonstrate how dynamic transitions between hidden states allow flexible reconfiguration of functional brain circuits. Crucially, we identify latent transient brain states and dynamic functional circuits that are optimal for cognition and show that failure to engage these states in a timely manner is associated with poorer task performance and weaker decision-making dynamics. We replicate findings in a large sample (N = 122) and reveal a robust link between cognition and flexible latent brain state dynamics. Our study demonstrates the power of switching dynamical systems models for investigating hidden dynamic brain states and functional interactions underlying human cognition.

Second-Order Switching Time Optimization for Switched Dynamical Systems

Switching time optimization arises in finite-horizon optimal control for switched systems where, given a sequence of continuous dynamics, one minimizes a cost function with respect to the switching times. We propose an efficient method for computing the optimal switching times for switched linear and nonlinear systems. A novel second-order optimization algorithm is introduced where, at each iteration, the dynamics are linearized over an underlying time grid to compute the cost function, the gradient and the Hessian efficiently. With the proposed method, the most expensive operations at each iteration are shared between the cost function and its derivatives, thereby greatly reducing the computational burden. We implemented the algorithm in the Julia package SwitchTimeOpt allowing the user to easily solve switching time optimization problems. In the case of linear dynamics, many operations can be further simplified and benchmarks show that our approach is able to provide optimal solutions in just a few ms. In the case of nonlinear dynamics, two examples show that our method provides optimal solutions with up to two orders of magnitude time reductions over state-of-the-art approaches.