State Dependent Ricatti Equation Research Articles

Optimal Floquet Stationkeeping under the Relative Dynamics of the Three-Body Problem

Deep space missions, and particularly cislunar endeavors, are becoming a major field of interest for the space industry, including for the astrodynamics research community. While near-Earth missions may be completely covered by perturbed Keplerian dynamics, deep space missions require a different modeling approach, where multi-body gravitational interactions play a major role. To this end, the Restricted Three-Body Problem stands out as an insightful first modeling strategy for early mission design purposes, retaining major dynamical transport structures while still being relatively simple. Dynamical Systems Theory and classical Hamiltonian Mechanics have proven themselves as remarkable tools to analyze deep-space missions within this context, with applications ranging from ballistic capture trajectory design to stationkeeping. In this work, based on this premise, a Hamiltonian derivation of the Restricted Three-Body Problem co-orbital dynamics between two spacecraft is introduced in detail. Thanks to the analytical and numerical models derived, connections between the relative and classical Keplerian and CR3BP problems are shown to exist, including first-order linear solutions and an inherited Hamiltonian normal form. The analytical linear and higher-order models derived allow the theoretical finding and unveiling of natural co-orbital phase space structures, including relative periodic and quasi-periodic orbital families, which are further exploited for general proximity operation applications. In particular, a novel reduced-order, optimal low-thrust stationkeeping controller is derived in the relative Floquet phase space, hybridizing the classical State Dependent Ricatti Equation (SDRE) with Koopman control techniques for efficient unstable manifold regulation. The proposed algorithm is demonstrated and validated within several end-to-end low-cost stationkeeping missions, and comparison against classical continuous stationkeeping algorithms presented in the literature is also addressed to reveal its enhanced performance. Finally, conclusions and open lines of research are discussed.

Adaptive inverse optimal backstepping control strategy for longitudinal vibration of high-speed elevator system based on fuzzy observer

To effectively suppress the longitudinal vibration of the car under the conditions of high-speed operation and emergency braking, and improve the ride comfort of the car system, this paper proposes an adaptive fuzzy inverse optimal output feedback control strategy. Firstly, the dynamic model of the high-speed elevator system is established and the nonlinear dynamic model is approximated by the fuzzy logic system, and the auxiliary system model is established. Fuzzy state observer is designed to estimate the unmeasurable state. Furthermore, an adaptive inverse optimal output feedback controller based on fuzzy observer is designed by using adaptive backstepping technology and inverse optimal control principle. The stability analysis shows that the proposed adaptive fuzzy inverse optimal output feedback control strategy not only ensures the stability of the car attitude of high-speed elevator but also realizes the inverse optimization of the target cost function. Finally, the acceleration time–frequency response analysis of the two typical stages of high-speed elevator uniform running and emergency braking is carried out, and the numerical results are compared with the linear quadratic controller optimized by stepping quantum genetic algorithm (GA-LQR) and controller based on the state-dependent Ricatti equation (SDRE). The analysis verifies the effectiveness of the controller.

Laboratory Study of Microsatellite Control Algorithms Performance for Active Space Debris Removal Using UAV Mock-Ups on a Planar Air-Bearing Test Bed

In this paper, a planar air-bearing test bed with unmanned aerial vehicles (UAV) was used to test a microsatellite motion control system. The UAV mock-ups were controlled by four ventilator actuators that imitated the satellite thrusters and provided the required acceleration vector in the horizontal plane, and torque along the vertical direction. The mock-ups moved almost without friction along the planar air-bearing test bed due to the air cushion between the test bed surface and the flat mock-up base. The motion of the mock-ups motion imitated the motion of satellites in the orbital plane. The problem of space debris can be solved using special microsatellite missions able to dock to space debris objects and change their orbit. In this paper, two control algorithms based on the virtual potentials approach and the State Dependent Ricatti Equation (SDRE) controller, were proposed for docking to a non-cooperative space debris object. The algorithms were tested in a laboratory facility, and the results are presented and analyzed, including their main features demonstrated during the laboratory study. It was shown that the SDRE-based control was faster, although the virtual potential-based control required less characteristic velocity.

Nonlinear dynamics, chaos and control of the Hindmarsh-Rose neuron model

Mathematics has changed over time to comprise interdisciplinary fields of research, and considering this, biomathematics has arisen as an interface study. In this work, we analyze the dynamical behavior of the Hindmarsh-Rose (HR) neuron model, which describes the neuronal bursting in a single neuron. A stability study through the Lyapynov exponents method is proposed and evidence of a chaotic dynamics is presented. Therefore, a control design based on the State-Dependent Ricatti Equation (SDRE) is proposed aiming to reduce the oscillation of the system to a desired orbit. The results show that the controller is efficient and robust as a method for preventing epileptic seizures.

SDRE-Based Integral Sliding Mode Control for Wind Energy Conversion Systems

This paper proposes a novel integral sliding mode control (ISMC) scheme based on numerically solving a state-dependent Ricatti equation (SDRE), nonlinear feedback control for wind energy conversion systems (WECSs) with permanent magnet synchronous generators (PMSGs). Unlike the conventional ISMC, the proposed control system is designed with nonlinear near optimal feedback control part to take into account nonlinearities of the WECSs. The Taylor series are used to approximate the solutions of SDRE. More specifically, the nonlinear optimal feedback control has been obtained by solving continuous algebraic Ricatti and Lyapunov equations. Sliding variables are designed such that reaching phase is eliminated and stability is guaranteed. The proposed control method equipped with high-order observer can guarantee more superior results than linear techniques such as linear quadratic regulator (LQR), conventional ISMC, and first-order sliding-mode control (SMC) method. Increasing the number of terms of the Taylor's series of the proposed control law provides better approximation, therefore the performance is improved. However, this increases the computational burden. The effectiveness of the control method is validated via simulations in MATLAB/Simulink under nominal parameters and model uncertainties.

A supplemental treatment for chemotherapy: Control simulation using a mathematical model with estimated parameters based on in vivo experiment

Abstract: Combinations of experimental studies and theoretical models have always played a key role in the clinical development of cancer treatments. In this study, in-vivo experiments are conducted in order to study the effects of DDW on melanoma tumor growth and its pharmacokinetics. Existing mathematical models are then employed to describe the melanoma dynamics and drug effects on the tumor growth. The model includes the tumor growth, chemotherapy, and pharmacokinetics. The model parameters are computed by fitting the mathematical relations to the test data using genetic algorithm (GA). A SDRE (State Dependent Ricatti Equation) based control strategy is then synthetized that leads to promising results. An optimal treatment plan is obtained consisting chemotherapy and DDW as a supplement treatment.

Direct Yaw Control of Vehicle using State Dependent Riccati Equation with Integral Terms

Direct yaw control of four-wheel vehicles using optimal controllers such as the linear quadratic regulator (LQR) and the sliding mode controller (SMC) either considers only certain parameters constant in the nonlinear equations of vehicle model or totally neglect their effects to obtain simplified models, resulting in loss of states for the system. In this paper, a modified state-dependent Ricatti equation method obtained by the simplification of the vehicle model is proposed. This method overcomes the problem of the lost states by including state integrals. The results of the proposed system are compared with the sliding mode slip controller and statedependent Ricatti equation method using high fidelity vehicle model in the vehicle simulation software package, Carsim. Results show 38% reduction in the lateral velocity, 34% reduction in roll and 16% reduction in excessive yaw by only increasing the fuel consumption by 6.07%.

Preventing Chaotic Motion in Tapping-Mode Atomic Force Microscope

During the last 30 years the Atomic Force Microscopy became the most powerful tool for surface probing in atomic scale. The Tapping-Mode Atomic Force Microscope is used to generate high quality accurate images of the samples surface. However, in this mode of operation the microcantilever frequently presents chaotic motion due to the nonlinear characteristics of the tip-sample forces interactions, degrading the image quality. This kind of irregular motion must be avoided by the control system. In this work, the tip-sample interaction is modelled considering the Lennard-Jones potentials and the two-term Galerkin aproximation. Additionally, the State Dependent Ricatti Equation and Time-Delayed Feedback Control techniques are used in order to force the Tapping-Mode Atomic Force Microscope system motion to a periodic orbit, preventing the microcantilever chaotic motion.

On linear and nonlinear trajectory tracking control for nonholonomic integrator

This paper presents two different kinds of trajectory tracking control strategies for the nonholonomic integrator known in literature as Brockett system. The first strategy presents a time-varying linear feedback control law and the second strategy is based on State Dependent Ricatti Equation (SDRE) method. Numerical simulation results indicated that both methods can be successfully used for control of the nonholomic integrator.Keywords: Brockett integrator, Hamilton – Jacobi – Bellman equation, SDRE method.

Nonlinear control in an electromechanical transducer with chaotic behaviour

The electromechanical transducer considered in this work is composed of a mechanical oscillator linked to an electronic circuit. Simulations results have determined that for some combination of parameters the electromechanical system is subject to chaotic motion with resonant transient behavior, and after the resonant transient the mechanical system (MS) synchronizes with the electrical system (ES). In order to improve the transient response, avoiding both the chaotic and resonant behaviors, a nonlinear control system is designed, a feedback control strategy is used to drive the system into the desired periodic orbit, and a nonlinear feedforward strategy is used to keep the system into the periodic orbit, obtained by the Fourier series. Two control techniques are used and compared, namely: the state dependent Ricatti equation and the optimal linear feedback control. Numerical simulations results are shown in order to compare the results, considering parametric uncertainties. Additionally, the energy transfer “pumping” between the ES and the MS is also analysed.

Nonlinear Output Tracking and Disturbance Rejection for Autonomous Close-Range Rendezvous and Docking of Spacecraft

A new control scheme of nonlinear output tracking and disturbance rejection for autonomous close-range rendezvous and docking of spacecraft is proposed in a closed-loop structure. The robustness of the proposed control scheme is achieved through the sum of output-feedback control and state-feedback control as a closed-loop control structure. The designs of both control laws are based on full, 6-degrees-of-freedom, nonlinear dynamic models, which are transformed into a linear-like form using a state dependent Ricatti equation technique. The proposed control scheme ensures closerange rendezvous and meets the state tracking conditions for docking such that the convergence of state tracking errors should be ensured in the presence of unknown disturbances and uncertainties of the system parameters. Compared with the conventional state-dependent Ricatti equation method, the proposed control scheme yields superior state tracking performance and robustness in the presence of external disturbances and uncertain system parameters. The 6-degreesof-freedom numerical simulation results are presented to verify the effectiveness and capability of the proposed control scheme for autonomous close-range rendezvous and docking.

Nonlinear dynamics and control strategies: On a energy harvester vibrating system with a linear form to non-ideal motor torquet

In this paper, we deal with the research of a vibrating model of an energy harvester device, including the nonlinearities in the model of the piezoelectric coupling and the non-ideal excitation. We show, using numerical simulations, in the analysis of the dynamic responses, that the harvested power is influenced by non-linear vibrations of the structure. Chaotic behavior was also observed, causing of the loss of energy throughout the simulation time. Using a perturbation technique, we find an approximate analytical solution for the non-ideal system. Then, we apply both two control techniques, to keep the considered system, into a stable condition. Both the State Dependent Ricatti Equation (SDRE) control as the feedback control by changing the energy of the oscillator, were efficient in controlling of the considered non-ideal system.

Robust Position and Attitude Control for Spacecraft Formation Flying

AbstractAn integrated approach combining a state-dependent Ricatti equation (SDRE) and an extended Kalman filter (EKF) is applied to spacecraft formation flying for robust orbital and attitude maneuvers. The formation flying considered in this study is a two-spacecraft formation with a bounded out-of-orbit plane relative motion. The chaser is required to simultaneously perform large position and angle maneuvers with sufficient accuracy. The chaser is then required to maintain a relative orbit expressed in the local-vertical-local-horizontal frame with respect to the target and align its attitude with the target attitude for more than one orbital period. The controls of the chaser are formulated as a nonlinear optimal regulator problem using their highly nonlinear dynamics. To test the robustness of this integrated approach, highly nonlinear dynamics, including external disturbances, are employed as the truth plant that is then used to generate the measurements for an EKF. The tracking error is then comput...