State-dependent Coefficient Research Articles

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

Poles, shocks, and tygers: The time-reversible Burgers equation.

We construct a formally time-reversible, one-dimensional forced Burgers equationby imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equationand can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equationare equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.

Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches

Permanent Magnet Synchronous Motors (PMSMs) with high energy efficiency, reliable performance, and a relatively simple structure are widely utilised in various applications. In this paper, a suboptimal controller is proposed for PMSMs without sensors based on the state-dependent Riccati equation (SDRE) technique combined with customised impulsive observers (IOs). Here, the SDRE technique facilitates a pseudo-linearised display of the motor with state-dependent coefficients (SDCs) while preserving all its nonlinear features. Considering the risk of non-available/non-measurable states in the motor due to sensor and instrumentation costs, the SDRE is combined with IOs to estimate the PMSM speed and position states. Customised IOs are proven to be capable of obtaining quality, continuous estimates of the motor states despite the discrete format of the output signals. The simulation results in this work illustrate an accurate state estimation and control mechanism for the speed of the PMSM in the presence of load torque disturbances and reference speed changes. It is clearly shown that the SDRE-IO design is superior compared to the most popular existing regulators in the literature for sensorless speed control.

Optimal quadratic full state feedback control of nonlinear affine systems

The solution of the optimal quadratic full-state feedback closed-loop control of a nonlinear affine system problem is presented and solved. This solution uses the State-Dependent Coefficient (SDC) form of a nonlinear system and the Calculus of Variations. Further, it is shown that this solution enables the computation of the value function associated with the respective Hamilton-Jacobi-Bellman equation thus satisfying the necessary and sufficient conditions for optimality. The use of the SDC form representation in the optimal solution for a nonlinear affine system results in an explicit and exact full-state feedback closed-loop control implementation as opposed to the existing open-loop solutions of the optimal control. This solution, like for the linear case, is non-causal and thus cannot be solved in real-time. The closed-loop application involves precomputing a time-varying full-state feedback matrix. The optimal feedback is a linear combination of the current state. The time-varying gain matrix is the backward solution of a non-symmetric matrix Riccati equation. Conditions for the stability of the full-state feedback are presented. This new representation of the optimal solution gives a well-understandable algorithm that enables implementation in the closed loop of the optimal feedback control of a non-linear system which is not possible with the existing optimal open loop representations.

An approximating state-dependent control method based on modified pattern search optimization for nonlinear optimal control problem

In this paper, an approximating state-dependent control (ASC) method with modified pattern search (MPS) optimization for nonlinear optimal control problem is proposed. First, by converting the nonlinear optimal control problem into a number of interrelated time-varying linear quadratic regulator subproblems, the ASC method can solve each subproblem iteratively until the approximate solution is obtained. Second, in each iterative control process, the MPS is used to solve the controllability optimization problem. The optimal state-dependent weighting coefficients are obtained during the MPS optimization. Moreover, the MPS uses simplex gradient to design the search direction, which makes the optimization process efficient and fast. The convergence of MPS optimization is also proved in this paper. Finally, two simulation examples are given to illustrate the effectiveness of ASC method using the MPS optimization. The result shows that the ASC method can reduce the iterations of the approximate solution, and the MPS optimization can optimize the control performance of the ASC method.

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation.

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Integrated guidance and control with impact angle and general field-of-view constraints

This paper investigates the problem of integrated guidance and control (IGC) law design with impact angle and general field-of-view (FOV) constraints. Firstly, the IGC model for intercepting non-maneuvering moving target is parameterized by state-dependent coefficient matrices. The nominal IGC law for intercepting the target with the desired impact angle is obtained by solving the state-dependent Riccati equation. Secondly, since the relative degrees of the general FOV constraints with respect to the IGC model exceed one, the high-order control barrier functions are constructed. The satisfaction of the FOV constraints is equivalent to ensuring that the superlevel sets defined by the barrier functions are forward invariant, which is converted into affine constraints on the control input. The nominal IGC law is modified in a minimally invasive way by quadratic programming. Then, the proposed method is extended to the case of intercepting maneuvering target by leveraging a relative coordinate frame. Finally, numerical simulations are conducted to verify the effectiveness of the proposed method.

Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems

The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are controlled by the interaction with the slow system. Using the theory of convergence of gradient systems depending on a small parameter ɛ (here the ratio between the slow and the fast time scale) in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a so-called B-function where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reactions give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.

Fixed-time stability super-twisting sliding mode control with mismatched disturbances

In practice, many systems are affected by mismatched disturbances, which do not satisfy the match condition. This study focuses on a variable gain super-twisting sliding mode controller that achieves global robust fixed-time stabilization of second-order systems with matched and mismatched disturbances without disturbances observers. The proposed controller and the sliding variables adopt a state-dependent variable exponent coefficient, which improves the robustness properties of a sliding surface and switching control law. At the same time, the control strategy alleviates the chattering phenomenon by using the variable-gain super-twisting algorithm. The settling time is calculated a priori by the Lyapunov method. Simulation results show that, compared with recently published controllers, our proposed control strategy can suppress the mismatched disturbance more rapidly, simply, and effectively.

Head-on collision of large-scale high density plasmas jets: A first-principle kinetic simulation approach

In the double-cone ignition (DCI) inertial confinement fusion (ICF) scheme, head-on collision of high density plasma jets is one of the most distinguished feature when compared with the traditional central ignition and fast ignition of ICF. However, the application of traditional hydrodynamic simulation methods becomes limited, due to serious plasma penetrations, mixing, and kinetic physics that might occur in the collision process. To overcome such limitations, we propose a new simulation method for large-scale high density plasmas. This method takes advantages of modern particle-in-cell simulation techniques and binary Monte Carlo collisions, including both long-range collective electromagnetic fields and short-range particle–particle interactions. Especially, in this method, the restrictions of simulation grid size and time step, which usually appear in a fully kinetic description, are eliminated. In addition, collisional coupling and state-dependent coefficients, which are usually approximately used with different forms in fluid descriptions, are also removed in this method. Energy and momentum exchanges among particles and species, such as thermal conductions and frictions, are modeled by “first principles” kinetic approaches. The correctness and robustness of the new simulation method are verified, by comparing with fully kinetic simulations at small scales and purely hydrodynamic simulations at large scale. Following the conceptual design of the DCI scheme, the colliding process of two plasma jets with initial density of 100 g/cc, initial thermal temperature of 65 eV, and counter-propagating velocity at 300 km/s is investigated using this new simulation method. Quantitative values, including density increment, increased plasma temperature, confinement time at stagnation, and conversion efficiency from the colliding kinetic energy to thermal energy, are obtained with a density increment of about three times, plasma temperature of 400 eV, confinement time at stagnation of 50 ps, and conversion efficiency of 85%. These values agree with the recent experimental measurements at a reasonable range.

Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

Calibration of high-fidelity hydrodynamic models utilizing on-site vessel response measurements

With a focus on weather-restricted marine operations, a comprehensive work on numerical model calibration is conducted to improve the vessel response prediction accuracy in real environmental conditions. This framework involves using existing physics-based models in which the metocean and system parameters are entered as inputs to derive vessel responses. The numerical models based on linear potential flow theory resolve only a part of the physics, i.e., it does not account for viscous damping effects. Moreover, significant uncertainties are associated with the system parameters related to the vessel operational conditions that are not known or measured directly. Therefore, the present paper integrates the vessel model with a derivative-free Lipschitz optimization technique to calibrate the vessel system parameters. For efficient optimization, the most influential variables are identified by conducting a sensitivity study for relevant quantities of interest using a cost-effective Polynomial Chaos model. The sensitivity analysis assumes independent uniform distributions for the system parameters with proper lower and upper limits. Subsequently, estimates of the highly influential variables are optimized using on-site vessel measurements, and in turn, utilized for superior response estimation. The response computations based on the numerical wave spectrum produced superior results in comparison with the results based on the parametric wave spectrum. The calibration algorithm is validated with full-scale measurements using frequency and time domain approaches for simulating vessel motions. It has been quantitatively revealed that the Center of Gravity parameters associated with the vessel operational characteristics and sea state-dependent additional damping coefficients govern the Roll response variation to a great extent. The calibration of these parameters resulted in improved Roll spectra estimations that match the measured spectra closely.

Control Design for Dengue Fever Model with Disturbance

A mathematical model has become a useful tool to predict and control dengue fever dynamics. In reality, the dynamic of dengue fever transmission can be disturbed by uncertainty measurements, so it is needed to consider the disturbance in the model. Then, dengue fever model with disturbance is constructed by using a gain matrix consisting a covariance matrix and random vector. As dengue vaccine has been challenging to reduce the pandemic, a dengue model with vaccination as control is constructed. The aim is to propose a feedback controller that can reduces the infected human (H2 control problem) and the uncertainty measurements (H∞ control problem). The control u denotes the proportion of susceptible humans that one decides to vaccinate at time t. A random mass vaccination with wanning immunity is chosen because vaccine still on development process. A Design of mixed H2 - H∞ control with State-dependent Riccati Equation (SDRE) approach is applied. The SDRE has been an effective method to solve for synthesizing nonlinear feedback controller by transforming the system to an State-dependent coefficient (SDC) form. By comparing the mixed scheme with basic H∞, numerical simulation shows that the control application effectively decreases the number of infected humans and reduces the disturbance.

An Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

We consider the construction of a feedback according to the Kalman algorithm for a continuous nonlinear control system on a finite time interval with control constraints where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position. The solution of an auxiliary optimal control problem with a quadratic functional is used for this task by analogy with the SDRE approach. Because this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems where all coefficient matrices in differential equations and criteria can contain state variables, on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations with state-dependent coefficient matrices. Due to the nonlinearity of the system this issue significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy in comparison with the Kalman algorithm for linear-quadratic problems. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V.F. Krotov and developed by V.I. Gurman and allows one not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. This article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, which depend on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

Modeling and Estimation of Self-Phoretic Magnetic Janus Microrobot With Uncontrollable Inputs

This study theoretically investigates the modeling of spherical catalytic self-phoretic magnetic Janus microrobot (MJR) evolving in uniform viscous flow. A 2-D state-space representation of the MJR is developed, exhibiting a state-dependent-coefficient (SDC) form. To evaluate the consistency of the modeling formalism, a dual Kalman filter (DKF) methodology is employed with respect to experimental results when unknown parameters or uncontrollable inputs are considered. In fact, the self-phoretic thrust mechanism and the magnetodynamics of the MJR are not well-known. SDC-DKF is implemented, and we find that there is good agreement between the dynamics computed from our theoretical predictions and the experimental observations in a wide range of model parameter variations.

Corrections to “Set-Membership Filter for Discrete-Time Nonlinear Systems Using State-Dependent Coefficient Parameterization” [Feb 22 894-901

Some of the final sets of corrections requested in the proof did not unfortunately make it to the final version in [1]. We would like to point those out in this document.

Cooperative Control of Underwater Vehicle-Manipulator Systems Based on the SDC Method.

The paper considers the problem of cooperative control synthesis for a complex of underwater vehicle–manipulator systems (UVMS) to perform the work of moving a cargo along a given trajectory. Here, we used the approach based on the representation of nonlinear dynamics models in the form of state space with state-dependent coefficients (SDC-form). That allowed us to apply methods of suboptimal control with feedback based on the state-dependent differential Riccati equation (SDDRE) solution at a finite time interval, providing the change in control intensity with the transient effect of the system matrices in SDC form. The paper reveals two approaches to system implementation: a general controller for the whole system and a set of independent subcontrollers for UVMSs. The results of both approaches are similar; however, for the systems with a small number of manipulators, the common structure is recommended, and for the systems with a large number of manipulators, the approach with independent subcontrollers may be more acceptable. The proposed method of cooperative control was tested on the task of cooperative control for two UVMSs with six-link manipulators Orion 7R. The simulation results are presented in the article and show the effectiveness of the proposed method.

ИДЕНТИФИКАЦИЯ ДЕФЕКТОВ В ПРИВОДАХ НА ОСНОВЕ МЕТОДОВ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ

The paper exams the problem of identifying faults in the drives of robotic systems, the dynamicsmodel of which is described by linear differential equations. It is proposed to search for asolution to the fault identification problem based on the solution of an auxiliary optimal controlproblem for a dynamical system in which the role of an unknown vector function describingemerging faults is performed by some auxiliary control, which should provide a minimum for theresidual functional. Based on the solution of the auxiliary optimal control problem, a fault diagnosticobserver is proposed. In this case, the fault itself is found through the solution of the correspondingalgebraic Riccati equation and the differential equation for the auxiliary variable. Unlikepopular approaches to solving the problem of fault identification based on observers operatingin a sliding mode, the proposed method allows us to expand the class of systems for which theidentification problem can be solved. It is known that the methods of sliding mode observers designimpose certain restrictions on the systems under consideration. The proposed approach basedon optimal control can also give results for systems with nonlinear dynamics. In this case, methodsof approximate solution of optimal control problems based on the representation of the system inlinear form with state-dependent coefficients (the so-called State-dependent Riccati Equation,SDRE) are likely to be effective. The improvement of the proposed method in this direction will bethe subject of further research. The stated theory is shown on the example of fault identification ina DC drive. Different cases are considered for a system with complete observations (the entirestate vector is known) and with incomplete observations. It was shown during the simulation thatthe quality of faults identification can be improved by selecting the appropriate values of the penaltymatrices in the residual functional, while it is possible to achieve good diagnostics separatelythrough various channels of faults occurrence. The paper presents recommendations on the choiceof penalty matrices. The simulation results confirmed the operability of the diagnostic observerssynthesized using the proposed method.

SDC-based dynamic inversion controller for a fixed-wing aircraft

This work presents dynamic model inversion controller which uses state dependent coefficient matrices, and is applied for a fixed-wing aircraft attitude and velocity control problem. Numerical validation of the proposed controller is done using a Cessna 172 aircraft simulation model for several scenarios of atmospheric disturbances.

Dynamic positioning of remotely operated vessel based on novel analysis and design of SDRE scheme

This article presents a new analysis and design for the dynamic positioning of remotely operated vessel using the state-dependent Riccati equation (SDRE) scheme. The focus is on the computational performance for efficient communication. The novel analysis completely removes the dominant routine (applicability/solvability check) at each time instant or system state, which differs from the common numerical compromise that is subject to limited states and precision. The design framework also facilitates the analysis of the α-parameterization technique, which is a usual practice (built on empirical confidence) but, actually, cannot guarantee alleviation of computational burden. The finding, in a unified manner, clarifies whether the practice is effective with respect to the associated eight degrees of freedom, which separate into two respective groups by virtue of a simple condition with reduced dimensionality. At those newly discovered system states corresponding to unsolvable SDREs, we alternatively introduce a solution to construct feasible state-dependent coefficients (SDCs), and maximize/unify its general capability with a proof and a MATLAB® implementation. From a practical viewpoint, numerical experiments also validate the proposed combined scheme, which entails (i) computationally enhanced classical SDC construction with verification and (ii) an efficient alternative. Finally, we demonstrate the impact within and beyond this marine system.