State-dependent Riccati Equation Filter Research Articles

Design and Validation of a State-Dependent Riccati Equation Filter for State of Charge Estimation in a Latent Thermal Storage Device

Abstract Latent thermal energy storage (TES) devices could enable advances in many thermal management applications, including peak load shifting for reducing energy demand and cost of HVAC or providing supplemental heat rejection in transient thermal management systems. However, real-time feedback control of such devices is currently limited by the absence of suitable state of charge estimation techniques, given the nonlinearities associated with phase change dynamics. In this paper, we design and experimentally validate a state-dependent Riccati equation (SDRE) filter for state of charge estimation in a phase change material (PCM)-based TES device integrated into a single-phase thermal-fluid loop. The advantage of the SDRE filter is that it does not require linearization of the nonlinear finite volume model; instead, it uses a linear parameter-varying system model which can be quickly derived using graph-based methods. We leverage graph-based methods to prove that the system model is uniformly detectable, guaranteeing that the state estimates are bounded. Using measurements from five thermocouples embedded in the PCM of the TES and two thermocouples measuring the fluid temperature at the inlet and outlet of the device, the state estimator uses a reduced-order finite volume model to determine the temperature distribution inside the PCM and in turn, the state of charge of the device. We demonstrate the state estimator in simulation and on experimental data collected from a thermal management system testbed to show that the state estimation error converges near zero and remains bounded.

Nonlinear State Dependent Riccati Equation Controller for 3-DOF Airfoil with Cubic Structural Nonlinearity

In this paper, Limit Cycle Oscillations (LCOs) of a Two-Dimensional (2-D) airfoil with cubic structural nonlinearity in the plunge Degree of Freedom (DOF) are investigated. The aerodynamic loads are modelled using the unsteady Theodorsen’s theory. Wagner function and Jones’ approximation are used to transform the unsteady aerodynamic loads from frequency domain into time domain. The aeroelastic differential equations are solved using the routine ODE45 in MATLAB to get the system response. Both subcritical and supercritical LCOs are observed in the 3-DOF airfoil with structural nonlinearity in the plunge DOF. Such LCOs are undesirable phenomena and should be suppressed within the flight envelope. A nonlinear state feedback controller is designed to minimize the amplitude of LCOs presented in the considered nonlinear aeroelastic system. The State-Dependent Riccati Equation (SDRE) approach in combination with Kalman filter technique is used to design a controller for the 2-D airfoil with trailing edge control surface (flap). The forces and moments produced by this flap’s action are used to stabilize the system and suppress the existence of LCOs. The efficiency of the proposed nonlinear controller by the SDRE and Kalman filter in suppressing the existence of subcritical and supercritical LCOs are verified by the numerical simulation results.

A novel frequency tracker for sinusoidal signal based on state dependent Riccati Equation filter

This paper presents a new approach for frequency estimation and tracking based on State Dependent Riccati Equation Filter (SDREF). The SDREF is an alternative concept to the well-known Extended Kalman Filter (EKF) to design a robust nonlinear filter, with low computational costs and better performances. The aim is to estimate and track the time-varying fundamental frequency of harmonic sinusoidal signals in general and more particularly the fundamental electric frequency. Through this study, the performance of the proposed approach is compared to the widely adopted the EKF, taking into account several aspects both theoretically and real-life applications. It has been shown that the proposed estimator performs better than the usual EKF estimator in terms of noise robustness, estimation accuracy, and tracking ability both in numerical simulations and experimental study.

A framework for 3D tracking of frontal dynamic objects in autonomous cars

Both recognition and 3D tracking of frontal dynamic objects are crucial problems in an autonomous vehicle, while depth estimation as an essential issue becomes a challenging problem using a monocular camera. Since both camera and objects are moving, the issue can be formed as a structure from motion (SFM) problem. In this paper, to elicit features from an image, the YOLOv3 approach is utilized beside an OpenCV tracker. Subsequently, to obtain the lateral and longitudinal distances, a nonlinear SFM model is considered alongside a state-dependent Riccati equation (SDRE) filter and a newly developed observation model. Additionally, a switching method in the form of switching estimation error covariance is proposed to enhance the robust performance of the SDRE filter. The stability analysis of the presented filter is conducted on a class of discrete nonlinear systems. Furthermore, the ultimate bound of estimation error caused by model uncertainties is analytically obtained to investigate the switching significance. Simulations are reported to validate the performance of the switched SDRE filter. Finally, real-time experiments are performed through a multi-thread framework implemented on a Jetson TX2 board, while radar data is used for the evaluation.

Robust partly strong tracking consider SDRE filter for direct INS/GNSS integration with biases

In order to counteract the adverse effects of biases in direct inertial navigation system/global navigation satellite system integration, a novel robust partly strong tracking consider state-dependent Riccati equation filter (PSTCSDREF) algorithm is proposed. A nonlinear ‘consider’ approach is utilized to incorporate bias statistics into a state estimation error covariance of the state-dependent Riccati equation filter (SDREF), and a new consider SDREF (CSDREF) is proposed. Next, the prediction covariance of the state is partly multiplied by a strong tracking factor, which does not include bias covariance, so as to mitigate the adverse effects of model uncertainties on navigation performance. Numerical simulation demonstrates that the proposed PSTCSDREF demonstrates better navigational accuracy as compared with the extended Kalman filter, SDREF, and CSDREF.

A switched SDRE filter for state of charge estimation of lithium-ion batteries

Lithium-ion (Li-ion) batteries need very precise monitor of the state of charge (SOC) to ensure a long cycle life. Hence, a knowledge of the SOC is important for Li-ion batteries. Although SOC cannot be measured directly, it can be estimated from direct measurement variables based on a model of the battery. Single-Particle-Model (SPM), a reduced-order nonlinear electrochemical model, is commonly used for this purpose. State-dependent-Riccati-equation (SDRE) filter is chosen as the estimator due to its high-flexibility in handling the model’s nonlinearity. However, performance of this filter is limited in presence of uncertainties. To tackle this problem, in this paper, a switching concept is induced into SDRE filter, in the form of switched estimation error covariance matrix with a certain frequency. Thus, by changing the Riccati equation dynamic in SDRE filter and proper adjustment of estimation error covariance matrix eigenvalues, performance and robustness of the common SDRE filter is significantly improved for Li-ion SOC estimation. To analyze the fidelity of such a filter in further applications, stability analysis is carried out on a class of nonlinear systems, and ultimate bound of estimation error is analytically obtained, and the influence of switching is investigated. Simulation results reveal effectiveness of the proposed filter compared to common SDRE filter, extended Kalman filter and variable structure approaches. Furthermore, experimental results verify the effectiveness of the proposed method in practice.

Development of a novel nonlinear estimator based on state-dependent Riccati equation technique for articulated vehicles

In this article, for the first time, a novel nonlinear estimator based on state-dependent Riccati equation filter technique is developed for state estimation of the articulated heavy vehicles. The state-dependent Riccati equation filter approach has a structure similar to the Kalman filter, and compared to the Kalman filter, which is based on linearization, the state-dependent Riccati equation filter is based on parameterization. Also, the state-dependent Riccati equation approach due to the nonuniqueness of the state-dependent coefficient matrix prevented singularity (uncontrollability or unobservability), and this advantage can be used to improve the efficiency of the system. For this purpose, using the Newton's method, a ten-degrees-of-freedom nonlinear model of the articulated heavy vehicle including the longitudinal, lateral and yaw motion of the tractor, the articulation angle, and rotational motion of each wheel is developed. Then, the developed model of articulated heavy vehicle is verified using nonlinear TruckSim model in high-speed lane change maneuver. The vehicle model validation results indicated that the development model is near to the actual articulated vehicle nonlinear model and can be utilized in the nonlinear estimator design. Then, using the state-dependent Riccati equation filter approach, the state variable estimation algorithm based on parametrization is designed and the articulated vehicle states are estimated online. In order to assess the performance and the efficiency of the developed estimator, the simulations with two standard maneuvers including low-speed 90 ° turn maneuver and high-speed lane change maneuver in the slippery road are performed. The simulation results indicate the remarkable impacts of the developed estimator based on state-dependent Riccati equation filter technique on the state estimation of the articulated heavy vehicles.

Relative navigation for autonomous formation flying satellites using the state-dependent Riccati equation filter

A relative navigation method for autonomous formation flying using the state-dependent Riccati equation filter (SDREF) is presented. In the SDREF, nonlinear relative dynamics, including J2 perturbation, are parameterized into a state-dependent coefficient (SDC) form without any loss of nonlinearity. The relative navigation algorithm is established based on the carrier-phase differential GPS (CDGPS) and single-frequency GPS data, in which the SDREF is used as a nonlinear estimator. To evaluate the SDREF performance, two different extended Kalman filters (EKFR1 and EKFR2) are introduced. The dynamic models of all the filters are based on relative motion including J2 perturbation. However, the SDREF and the EKFR1 use linear state propagation, whereas EKFR2 employs nonlinear state propagation. The navigation simulation is performed for each filter using live GPS signals simulated by a GPS signal generator, and the result is analyzed in terms of estimation accuracy and computational load. As a result, the SDREF provides a relative navigation solution with 3-D RMS accuracies of 6.0mm and 0.153mm/s for position and velocity, respectively, for a separation of 50km with a computation time of approximately 34s. The simulation results demonstrate that the SDREF estimates the relative states as rapidly as the EKFR1 and as accurately as the EKFR2, which means that the developed SDREF combines the strong points of EKFR1 and EKFR2 and overcomes their disadvantages.

SINS Transfer Alignment Based on Robust SDRE Filter

In order to improve the rapid response ability of the inertial navigation system mounted on the aircrafts which are onboard a ship, a novel rapid transfer alignment method of the strapdown inertial navigation system (SINS) on a rocking base is put forward.Interfering motions such as rolling, pitching, and yawing motions caused by sea waves are effectively used. Meanwhile, to remedy the effects of modeling uncertainty and adapt the nonlinear character in the error models, a robust SDRE (state-dependent Riccati equation) filter design is developed based on the infinity-norm minimization criterion. The filter presented can solve the problem that the standard Kalman filter cannot give the optimal solution when the system model and stochastic information are unknown accurately. Simulation results show that the misalignment angle error estimation can be upgraded dramatically and the new algorithm is less sensitive to uncertainty noise.

Nonlinear Finite-Horizon Regulation and Tracking for Systems with Incomplete State Information Using Differential State Dependent Riccati Equation

This paper presents an efficient online technique used for finite-horizon, nonlinear, stochastic, regulator, and tracking problems. This can be accomplished by the integration of the differential SDRE filter algorithm and the finite-horizon state dependent Riccati equation (SDRE) technique. Unlike the previous methods which deal with the linearized system, this technique provides finite-horizon estimation and control of the nonlinear stochastic systems. Further, the proposed technique is effective for a wide range of operating points. Simulation results of a missile guidance system are presented to illustrate the effectiveness of the proposed technique.

Design of Satellite Attitude Control Algorithm Based on the SDRE Method Using Gas Jets and Reaction Wheels

An experimental attitude control algorithm design using prototypes can minimize space mission costs by reducing the number of errors transmitted to the next phase of the project. The Space Mechanics and Control Division (DMC) of INPE is constructing a 3D simulator to supply the conditions for implementing and testing satellite control hardware and software. Satellite large angle maneuver makes the plant highly nonlinear and if the parameters of the system are not well determined, the plant can also present some level of uncertainty. As a result, controller designed by a linear control technique can have its performance and robustness degraded. In this paper the standard LQR linear controller and the SDRE controller associated with an SDRE filter are applied to design a controller for a nonlinear plant. The plant is similar to the DMC 3D satellite simulator where the unstructured uncertainties of the system are represented by process and measurements noise. In the sequel the State-Dependent Riccati Equation (SDRE) method is used to design and test an attitude control algorithm based on gas jets and reaction wheel torques to perform large angle maneuver in three axes. The SDRE controller design takes into account the effects of the plant nonlinearities and system noise which represents uncertainty. The SDRE controller performance and robustness are tested during the transition phase from angular velocity reductions to normal mode of operation with stringent pointing accuracy using a switching control algorithm based on minimum system energy. This work serves to validate the numerical simulator model and to verify the functionality of the control algorithm designed by the SDRE method.

Nonlinear Control of Formation Flying with State Constraints

Nonlinear Control of Formation Flying with State Constraints

Exponential nonlinear observer based on the differential state-dependent Riccati equation

This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation (SDRE) filter with guaranteed exponential stability. Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters, the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence. In this paper, Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics. We prove that under specific conditions, the proposed observer is at least locally exponentially stable. Moreover, a new definition of a detectable state-dependent factorization is introduced, and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established. Furthermore, through a simulation study of a second order nonlinear model, which satisfies the stability conditions, the promising performance of the proposed observer is demonstrated. Finally, in order to examine the effectiveness of the proposed method, it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines. The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate.

Application of the State-Dependent Riccati Equation and Kalman Filter Techniques to the Design of a Satellite Control System

Design of Satellite Attitude Control System (ACS) that involves plant uncertainties and large angle manoeuvres following a stringent pointing control, may require new non-linear control techniques in order to have adequate stability, good performance and robustness. In that context, experimental validation of new non-linear control techniques through prototypes is the way to increase confidence in the controller designed. The Space Mechanics and Control Division (DMC) of INPE is constructing a 3-D simulator to supply the conditions for implementing and testing satellite ACS hardware and software. The 3-D simulator can accommodate various satellites components; like sensors, actuators, computers and its respective interface and electronic. Depending on the manoeuvre the 3-D simulator plant can be highly non-linear and if the simulator inertia parameters are not well determined the plant also can present some kind of uncertainty. As a result, controller designed by linear control technique can have its performance and robustness degraded, therefore controllers designed by new non-linear approach must be considered. This paper presents the application of the State-Dependent Riccati Equation (SDRE) method in conjunction with Kalman filter technique to design a controller for the DMC 3-D satellite simulator. The SDRE can be considered as the non-linear counterpart of Linear Quadratic Regulator (LQR) control technique. Initially, a simple comparison between the LQR and SDRE controller is performed. After that, practical applications are presented to address problems like presence of noise in process and measurements and incomplete state information. Kalman filter is considered as state observer to address these issues. The effects of the plant non-linearities and noises (uncertainties) are considered in the performance and robustness of the controller designed by the SDRE and Kalman filter. The 3-D simulator simulink-based model has been developed to perform the simulations examples to investigate the SDRE controller performance using the states estimated by the Kalman filter. Simulations have demonstrated the validity of the proposed approach, once the SDRE controller has presented good stability margin, great performance and robustness.

Application of the State-Dependent Riccati Equation and Kalman Filter Techniques to the Design of a Satellite Control System

Design of Satellite Attitude Control System (ACS) that involves plant uncertainties and large angle manoeuvres following a stringent pointing control, may require new non-linear control techniques in order to have adequate stability, good performance and robustness. In that context, experimental validation of new non-linear control techniques through prototypes is the way to increase confidence in the controller designed. The Space Mechanics and Control Division (DMC) of INPE is con- structing a 3-D simulator to supply the conditions for implementing and testing satellite ACS hardware and software. The 3-D simulator can accommodate various satellites components; like sensors, actuators, computers and its respective interface and electronic. Depending on the manoeuvre the 3-D simulator plant can be highly non-linear and if the simulator inertia parameters are not well determined the plant also can present some kind of uncertainty. As a result, controller designed by linear control technique can have its performance and robustness degraded, therefore controllers designed by new non-linear approach must be considered. This paper presents the application of the State-Dependent Riccati Equation (SDRE) method in conjunction with Kalman filter technique to design a controller for the DMC 3-D satellite simulator. The SDRE can be considered as the non-linear counterpart of Linear Quadratic Regulator (LQR) control technique. Initially, a simple comparison between the LQR and SDRE controller is performed. After that, practical applications are presented to address problems like presence of noise in process and measurements and incomplete state information. Kalman filter is considered as state observer to address these issues. The effects of the plant non-linearities and noises (uncertainties) are considered in the performance and robustness of the controller designed by the SDRE and Kalman filter. The 3-D simulator simulink-based model has been developed to perform the simulations ex- amples to investigate the SDRE controller performance using the states estimated by the Kalman filter. Simulations have demonstrated the validity of the proposed approach, once the SDRE controller has presented good stability margin, great per- formance and robustness.

Robust SDRE filter design for nonlinear uncertain systems with an [formula omitted] performance criterion

In order to remedy the effects of modeling uncertainty, measurement noise and input disturbance on the performance of the standard state-dependent Riccati equation (SDRE) filter, a new robust H ∞ SDRE filter design is developed in this paper. Based on the infinity-norm minimization criterion, the proposed filter effectively estimates the states of nonlinear uncertain system exposed to unknown disturbance inputs. Moreover, by assuming a mild Lipschitz condition on the chosen state-dependent coefficient form, fulfillment of a modified H ∞ performance index is guaranteed in the proposed filter. The effectiveness of the robust SDRE filter is demonstrated through numerical simulations where it brilliantly outperforms the conventional SDRE filter in presence of model uncertainties, disturbance and measurement noise, in terms of estimation error and region of convergence.

Robust INS/GPS Sensor Fusion for UAV Localization Using SDRE Nonlinear Filtering

The aim of this paper is to present a new INS/GPS sensor fusion scheme, based on state-dependent Riccati equation (SDRE) nonlinear filtering, for unmanned aerial vehicle (UAV) localization problem. SDRE navigation filter is proposed as an alternative to extended Kalman filter (EKF), which has been largely used in the literature. Based on optimal control theory, SDRE filter solves issues linked with EKF filter such as linearization errors, which severely decrease UAV localization performances. Stability proof of SDRE nonlinear filter is also presented and validated on a 3-D UAV flight scenario. Results obtained by SDRE navigation filter were compared to EKF navigation filter results. This comparison shows better UAV localization performance using SDRE filter. The suitability of the SDRE navigation filter over an unscented Kalman navigation filter for highly nonlinear UAV flights is also demonstrated.

Asymptotically Optimal Nonlinear Filtering: Theory and Examples with Application to Target State Estimation

The State-Dependent Riccati Equation (SDRE) filter, which is derived by constructing the dual of the well-known SDRE nonlinear regulator control design technique, has been studied in various papers, with mainly practical investigations of the filter. Until recently, theoretical aspects of the filter had not been fully investigated, leaving many unanswered questions, such as stability and convergence of the filter. The authors conducted an investigation of the conditions under which the state estimate given by this algorithm converges asymptotically to the first order minimum variance estimate given by the extended Kalman filter (EKF). Conditions for determining a region of stability for the SDRE filter were also investigated. The analysis was based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. In this paper, the motivation for introducing HJB equations is justified with mathematical rigor, which is given by reference to the maximum likelihood approach to deriving the EKF. The application of the SDRE filter is then demonstrated on challenging examples to illustrate the theoretical aspects and design flexibility (additional degrees of freedom) of the algorithm when loss of observability is encountered. In particular, a realistic and detailed evaluation of the filter is carried out for the problem of target state estimation in an advanced tactical missile guidance application for analysis in the optimal guidance problem for air-air engagements using only passive sensor (angle-only) information. Simulation results are presented which show dramatic tracking improvement using the SDRE target tracker.

ASYMPTOTICALLY OPTIMAL NONLINEAR FILTERING

In this paper, a theoretical investigation of the state-dependent Riccati equation (SDRE) filter is carried out, which is derived by constructing the dual of the well-known SDRE nonlinear regulator control design technique. The SDRE filter has been studied in various papers, with mainly practical investigations of the filter. However, the theoretical aspects of the filter have not been fully investigated and there remain many unanswered questions, such as stability and convergence of the filter. This paper investigates conditions under which the state estimate given by this algorithm converges asymptotically to the first order minimum variance estimate given by the extended Kalman filter (EKF). Conditions for determining a region of stability for the SDRE filter are also investigated. The analysis is based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. The motivation for introducing HJB equations is given by reference to the maximum likelihood approach to deriving the EKF. The application of the SDRE filter will be demonstrated on a simple pendulum problem to illustrate the theory. The behavioral differences and similarities between the SDRE filter, the linearized Kalman filter (LKF) and the EKF are also discussed using this example.