State-dependent Riccati Equation Control Research Articles

Closed-loop nonlinear optimal control design for flapping-wing flying robot (1.6 m wingspan) in indoor confined space: Prototyping, modeling, simulation, and experiment

The flapping-wing technology has emerged recently in the application of unmanned aerial robotics for autonomous flight, control, inspection, monitoring, and manipulation. Despite the advances in applications and outdoor manual flights (open-loop control), closed-loop control is yet to be investigated. This work presents a nonlinear optimal closed-loop control design via the state-dependent Riccati equation (SDRE) for a flapping-wing flying robot (FWFR). Considering that the dynamic modeling of the flapping-wing robot is complex, a proper model for the implementation of nonlinear control methods is demanded. This work proposes an alternative approach to deliver an equivalent dynamic for the translation of the system and a simplified model for orientation, to find equivalent dynamics for the whole system. The objective is to see the effect of flapping (periodic oscillation) on behavior through a simple model in simulation. Then the SDRE controller is applied to the derived model and implemented in simulations and experiments. The robot bird is a 1.6 m wingspan flapping-wing system (six-degree-of-freedom robot) with four actuators, three in the tail, and one as the flapping input. The underactuated system has been controlled successfully in position and orientation. The control loop is closed by the motion capture system in the indoor test bed where the experiments of flight have been successfully done.

Synthesis of Suboptimal Guidance Law for Anti-Tank Guided Missile with Terminal Impact Angle Constraint Based on the SDRE Technique

In this paper, the homing-phase guidance law is proposed against a stationary target in the planar engagement scenario. In this problem, two technical criteria that the guidance law must achieve are: first, zero terminal miss distance, and second, satisfying the desired impact angle constraint at the final time. This guidance law is synthesized as a nonlinear optimal control problem with an infinite-time horizon and is solved using the state-dependent Riccati equation (SDRE) method. To obtain a guidance law with a finite-time horizon, a state weighting matrix based on the time-to-go is utilized in the SDRE control scheme. The synthesized guidance law is applied to a new generation anti-tank guided missile class, with specific thrust and drag parameters. Nonlinear simulations are conducted to demonstrate the fundamental properties, applicability, and effectiveness of the proposed guidance law.

Distilled neural state-dependent Riccati equation feedback controller for dynamic control of a cable-driven continuum robot

This article presents a novel learning-based optimal control approach for dynamic control of continuum robots. Working and interacting with a confined and unstructured environment, nonlinear coupling, and dynamic uncertainty are only some of the difficulties that make developing and implementing a continuum robot controller challenging. Due to the complexity of the control design process, a number of researchers have used simplified kinematics in the controller design. The nonlinear optimal control technique presented here is based on the state-dependent Riccati equation and developed with consideration of the dynamics of the continuum robot. To address the high computational demand of the state-dependent Riccati equation controller, the distilled neural technique is adopted to facilitate the real-time controller implementation. The efficiency of the control scheme with different neural networks is demonstrated using simulation results.

Time-energy suboptimal control of nonlinear input-affine systems

In this paper, we derive the necessary conditions for time-energy optimal control of input-affine nonlinear systems using Hamilton–Jacobi–Bellman equation. The objective function is chosen as a weighted combination of control effort and time-horizon minimisation. The necessary conditions of optimality are transformed such that the resultant conditions are structurally similar to a differential matrix Riccati equation. A closed-form approximate solution to these necessary conditions is derived such that the resulting closed-loop dynamics is locally exponentially stable around the origin. Furthermore, it is also shown that the states converge within a ball of an arbitrarily small radius around the origin within a finite time. Finally, the performance of the proposed controller is tested using numerical simulations for Vander Pol's oscillator as well as spacecraft detumbling problem. Furthermore, the performance of the proposed controller is also tested in comparison to the infinite-horizon state-dependent Riccati equation controller.

SDRE-based primary control of DC Microgrids equipped by a fault detection/isolation mechanism

Due to the nonlinear dynamics of direct current (DC) microgrids, the existence of input constraints, and their multi-input multi-output (MIMO) nature, classical linear controllers cannot provide an appropriate performance in a wide range of operations. In this paper, to address these issues, nonlinear suboptimal controllers are systematically developed in the primary layer of DC microgrids by employing a state-dependent Riccati equation (SDRE) methodology. To this end, the whole complexities of the nonlinear dynamics and input constraints are considered in the design procedure of the proposed SDRE controllers. After designing the controllers, and for a fast yet effective fault detection/isolation, an artificial neural network (ANN) is trained to identify the closed-loop microgrid at its nominal condition. Then, the trained ANN is employed to design a fault detection/isolation mechanism. Simulation results of the developed SDRE control scheme augmented by the ANN-based fault detection/isolation mechanism demonstrate the merits of the proposed scheme.

Nonlinear Sub-optimal Control Design for Suppressing HIV Replication

Acquired Immunodeficiency Syndrome is a deadly viral disease caused by the Human Immunodeficiency Virus in vivo, and its purpose is to destroy the immune system of the human body. The disease does not currently have a definitive vaccine or treatment, but treatment with pharmaceutical interventions (antiretroviral therapy, or ART) can slow down the progression of HIV. Daily use of prophylaxis measures may also have serious side effects for the patient, so the dosage and regimen of drugs should be constantly controlled. The dynamic models formulated for HIV infection are nonlinear differential equations. Therefore, nonlinear optimal control methods can be effective in increasing the efficiency of treatment. In this study, a sub-optimal controller based on the state-dependent Riccati equation (SDRE) approach to the dynamic model of HIV is introduced. One of the advantages of the SDRE approach is that the nonlinear properties of the system are preserved in the design control procedure. Furthermore, the specific conditions of infected individuals can be considered via choosing appropriate coefficients in the cost function and limiting the amount of drug administered. In the procedure of control design, all the state variables must be available for feedback in order to use the SDRE controller. In this regard, the Extended Kalman Filter observer is also implemented. The effect of different weighting matrices on these states is examined. In addition, to assess the effectiveness of the proposed control strategy, the well-known performance indicator root mean square error is also considered. Numerical simulations confirm the high efficiency and flexibility of the proposed approach.

Hybrid Quasi-Optimal PID-SDRE Quadrotor Control

In the paper, a new cascade control system for an autonomous flight of an unmanned aerial vehicle (UAV) based on Proportional–Integral–Derivative (PID) and finite-time State-Dependent Riccati Equation (SDRE) control is proposed. The PID and SDRE controllers are used in a hybrid control system for precise control and stabilization, which is necessary to increase the drone’s flight stability and maneuver precision. The hybrid PID-SDRE control system proposed for the quadrotor model is quasi-optimal, since the suboptimal control algorithm for the UAV stabilization is used. The combination of the advantages of PID and SDRE control gives a significant improvement in the quality of control while maintaining the simplicity of the control system. Furthermore, the use of the suboptimal control technique provides the UAV attitude tracking in finite time. These remarks are drawn from a series of simulation tests conducted for the drone model.

Optimal consensus control of multi-agent systems based on the state-dependent riccati equation

This paper is concerned with optimal consensus control of a multi-agent system, leaderless or leader-follower control. The hierarchical state-dependent Riccati equation (SDRE) controller was the main framework. The Lyapunov-based method is used for solving time-varying systems. Not only positive definite solution of the related algebraic Riccati equation is realized by the SDRE method based on Lyapunov, but also the negative definite solution is realized. Furthermore, the leader-follower tracking structure is developed based on two nonlinear differential equations; the feedforward differential equation and the state-dependent Riccati differential equation, which consider the error in the cost function through the co-sate equation. The application of the proposed controller structure in controlling the lifting system of multiple 3D cranes is studied. The theoretical results have been verified by simulating the leaderless and leader-follower consensus control problem.

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.

Model-Based Nonlinear Feedback Controllers for Pressure Control of Soft Pneumatic Actuators Using On/Off Valves.

This article describes the application and comparison of three nonlinear feedback controllers for low-level control of soft actuators driven by a pressure source and single high-speed on/off solenoid valve. First, a mathematical model of the pneumatic system is established and the limitations of the open-loop system are evaluated. Next, a model of the pneumatic system is developed using Simscape Fluids to evaluate the performance of various control strategies. In this article, State-Dependent Riccati Equation control, sliding mode control, and feedback linearization are considered. To improve robustness to model uncertainties, the sliding mode and feedback linearization control strategies are augmented with integral action. The model of the pneumatic system is also used to develop a feedforward component, which is added to a PI controller with anti-windup. The simulation and experimental results demonstrate the effectiveness of the proposed controllers for pressure tracking.

Maximum power tracking of variable‐speed wind energy conversion systems based on a near‐optimal servomechanism control system

AbstractIn this article, integral servomechanism based state‐dependent Riccati equation (SDRE) nonlinear output feedback control for wind energy conversion systems (WECSs) has been proposed. The numerical off‐line solving the proposed SDRE control requires to formulate algebraic Riccati equation and algebraic Lyapunov equation. This method approximates the solution of SDRE with Taylor series expansions which are solved with MATLAB solvers. The maximum power point tracking is achieved by defining relation between the optimal angular shaft speed and maximum power providing optimal tip speed ratio in variable‐speed WECSs. The proposed control can significantly reduce the angular shaft speed errors of the permanent magnet synchronous generator (PMSG)‐based WECSs without use of disturbance observers to compensate model uncertainties, modeling errors and noise as in the conventional SDRE control method. The simulation results demonstrate the superior results under mean average errors and root mean square error evaluation methods of the angular shaft speed of PMSG. The performance of the proposed integral servomechanism based SDRE control system has been improved by 80.67% and 80.05% in both scenarios compared to the conventional SDRE control based on compensation technique with disturbance observers.

Fuzzy State-Dependent Riccati Equation (FSDRE) Control of the Reverse Osmosis Desalination System With Photovoltaic Power Supply

The two challenges facing human life are water and energy. Reverse osmosis (RO) desalination systems are popular owning to their unique advantages. However, robust performance and power supply are the two main challenges in this desalination system. This power is used to drive an induction motor that rotates a centrifugal pump to apply the required back pressure to the RO membrane. To solve these two challenges, a complete RO system powered by a photovoltaic (PV) system was considered, and for each subsystem, a robust controller was designed based on their dynamic models. A fuzzy controller optimized by the invasive weed algorithm (IWA) was designed to track the maximum power in the photovoltaic subsystem under different environmental conditions. A fuzzy-PID controller was used to control the motor-pump subsystem. Furthermore, it is focused on designing a robust controller with the ability to compensate for large set-point changes, reject external disturbances, and cope with parametric uncertainties, such as variations in feed water salinity. Hence, state-dependent Riccati equation control (SDRE) was used to control the reverse osmosis system. The simulation results for different scenarios show that the proposed controller performs well under different operating conditions and can remove the effects of disturbances on the system.

Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs

A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a warm start for the minimization of a loss function based on the residual of the HJB PDE. The combination of supervised learning and residual minimization avoids spurious solutions and mitigate the data inefficiency of a supervised learning-only approach. Numerical tests validate the different advantages of the proposed methodology.

SDRE and LQR Controls Comparison Applied in High-Performance Aircraft in a Longitudinal Flight

This paper presents the design of the LQR (Linear Quadratic Regulator) and SDRE (State-Dependent Riccati Equation) controllers for the flight control of the F-8 Crusader aircraft considering the nonlinear model of longitudinal movement of the aircraft. Numerical results and analysis demonstrate that the designed controllers can lead to significant improvements in the aircraft's performance, ensuring stability in a large range of attack angle situations. When applied in flight conditions with an angle of attack above the stall situation and influenced by the gust model, it was demonstrated that the LQR and SDRE controllers were able to smooth the flight response maintaining conditions in balance for an angle of attack up to 56% above stall angle. However, for even more difficult situations, with angles of attack up to 76% above the stall angle, only the SDRE controller proved to be efficient and reliable in recovering the aircraft to its stable flight configuration.

On global stability of planar systems with state‐dependent Riccati equation control

AbstractThis paper is concerned with the global asymptotic stability (GAS) of a class of planar systems with a state‐dependent Riccati equation (SDRE) control. Based on the philosophy of feedback domination along with the depiction of the influence arising from system nonlinearities, a sufficient condition indicating the allowable intervals of control gains is established by which the GAS of the system with SDRE control can be guaranteed successfully. The presented condition is delicately equipped with a tunable parameter skillfully enhancing the feasibility while providing a new insight and perspective in ensuring global stability of the SDRE control. A numerical example is provided to verify the effectiveness of the proposed result.

Fast real-time SDRE controllers using neural networks

This paper describes the implementation of fast state-dependent Riccati equation (SDRE) control algorithms through the use of shallow and deep artificial neural networks (ANN). Several ANNs are trained to replicate an SDRE controller developed for a satellite attitude dynamics simulator (SADS) to display the technique’s efficacy. The neural controllers have reduced computational complexity compared with the original SDRE controller, allowing its execution at a significantly higher rate. One of the neural controllers was validated using the SADS in a practical experiment. The experimental results indicate that the training error is sufficiently small for the neural controller to perform equivalently to the original SDRE controller.

State Dependent Riccati Equation (SDRE) controller design for moving obstacle avoidance in mobile robot

In this paper, a new navigating method for a non-holonomic wheeled moving robot in a dynamic environment with the moving obstacles is proposed. This method is indeed the combination of the nonlinear optimal controller based on State-Dependent Riccati Equation (SDRE) and an obstacle avoidance algorithm named artificial potential field (APF). The corresponding cost function of the SDRE is obtained of APF algorithm. APF algorithm forces the robot to approach the target as an attracting (low-potential) point and to get away from the obstacle as a repulsing (high-potential) point. This method calculates the best path from origin to destination which also implicitly guaranties the stability. The obtained path is the best according to the both amount of traveled distance and input energy. Moreover, this approach not only avoid both fixed and moving obstacle, but also create a smooth path in presence of them. Keeping the nonlinear structure of the system instead of eliminating them during the linearization process is the advantage of SDRE method. Here, the robot navigation is done in the presence of the three different movements of obstacle: (1) fixed speed, (2) fixed acceleration and (3) non-uniform circular. The represented simulation results indicate a suitable performance of the proposed algorithm.

Vibration suppression and attitude control for the formation flight of flexible satellites by optimally tuned on-off state-dependent Riccati equation approach

In this paper, vibration suppression and attitude control for the formation flight of flexible satellites using optimally tuned on-off SDRE (state-dependent Riccati equation) approach is discussed. A formation consisting of flexible satellites has highly nonlinear dynamics and the corresponding satellites are subject to vibrations as well as uncertainties due to the practical conditions. Vibrations that are mainly caused by flexible modes of the satellites disorganize the coordination and hinder the formation stability as well as decreasing its performance and lifetime. Hence, flexibility should be considered in formation model and the coordination control needs to address such challenges. Owing to capabilities of SDRE approach for nonlinear systems, it is used as the coordination control. Satellites are assumed to be equipped with thrusters as their actuators which requires the control to be applied as on-off pulses. To this end, an algorithm is suggested to efficiently convert SDRE control into on-off pulses. For optimal tuning of the controller, the particle swarm optimization (PSO) algorithm is employed. Stability of the system has also been analyzed via a Lyapunov-based approach utilizing the region of attraction concept. The proposed on-off SDRE approach has shown to effectively suppress the vibrations in the presence of uncertainties leading to the accurate coordination of the whole formation while consuming less energy. Simulation results show the capability, efficiency, robustness and stability of the suggested approach.

Vehicle tip-over prevention using gain-scheduled State-dependent Riccati Equation–based anti-rollover controller

The most hazardous kind of vehicle crash among all road accidents is vehicle rollover. Present-day rollover prevention systems in commercial vehicles mitigate rollover by preventing any wheel lift-off from the ground. These systems make use of actuators such as differential brakes and demand all the wheels on the ground for satisfactory operation. Such systems are not effective in recovering a vehicle from intense rollover scenarios where the wheels on one side are lifted off the ground, and the vehicle is about to rollover to the other side after reaching the tip-over point. A few studies have investigated the possibility of reinstating a vehicle at the tip-over point with the wheels on a side lifted off. The high complexity and computation time of the optimal control strategies such as nonlinear model predictive controller make it unsuitable for real-time implementations. This study proposes a novel gain-scheduled State-dependent Riccati Equation–based optimal anti-rollover controller for reinstating a vehicle from the tip-over point. An inverted double pendulum on a cart vehicle model is used as the plant model. The anti-rollover controller is found to be presentable as a two-dimensional gain-scheduled lookup table with specific state dependencies in existence. It eliminates the necessity of solving the nonlinear performance index minimization problem online. State-dependent Riccati Equation method adequately accounts for the nonlinearities involved, yet possesses a small computational time per sample. The anti-rollover controller is evaluated with a 10 degrees of freedom full vehicle model with a nonlinear pure slip tyre model that incorporates the dynamical effects neglected in the controller formulation. Finally, the anti-rollover controller is evaluated in real-life initial conditions using a sophisticated pick-up truck model obtained from TruckSim® software through a co-simulation with the anti-rollover controller setup in MATLAB®/Simulink® environment. The State-dependent Riccati Equation controller was found to be effective in reinstating the higher-order models from the tip-over point in all the case studies conducted.

Optimal Sliding Mode Controller for an Active Transfemoral Prosthesis Using State-Dependent Riccati Equation Approach

In this paper, a nonlinear robust optimal controller is designed for an active transfemoral prosthesis using sliding mode control and state-dependent Riccati equation control. In this method, unlike most nonlinear control methods, the Jacobian approach is not used for model linearization and system's state space equations are used directly. This is a reason for the accuracy and flexibility of the proposed controller in design compared to other conventional methods. Since the proposed approach is energy-driven, the main objective of this research is to optimize the energy consumption of the robot/prosthesis system, reduce the effects of disturbances, noise, and parametric uncertainties, and desirable trajectory tracking of the vertical displacement in hip and thigh and knee angles. To this aim, the integral state control technique and the boundary layer are used for reconciliation between tracking performance and control signal chattering. In this study, the performance of the controller is assessed for nominal system and uncertain system with ± 30% parameters’ uncertainty by considering the saturation bounds of control signals. The simulation results demonstrate a decrease in the control effort, good robustness in the presence of uncertainty, and desirable tracking performance compared to a robust adaptive impedance control approach.