State-dependent Riccati Equation Research Articles

Characterization and Verification of the Optimal Feedback Gain of a Satellite Magnetorquer-based Attitude Control System

In spacecraft mission planning and operation, the attitude determination and control subsystem (ADCS) of a satellite provides information about the orientation of the satellite in the inertial reference frame. Furthermore, this subsystem produces the control actions required to adjust the orientation of the satellite, especially in the low-Earth orbit (LEO) regime. This paper focuses on the satellite’s three-axis attitude control problem within the context of active and passive control, which includes detumbling control, pointing control, magnetic control, and attitude stabilization after solar panel wing deployment using magnetorquers as the primary actuators. The objective is to stabilize and reduce the angular rate while orienting the satellite to the desired attitude. The proposed satellite attitude control system (ACS) strategies are designed, developed, characterized, and verified. These strategies encompass the B-dot control algorithm for detumbling control along with pointing control and attitude stabilization after solar panel wing deployment. hardware-in-the-loop simulation (HiLs) tests are conducted to assess the performance of the satellite magnetorquer-based ACS in the presence of noise. These tests involve a relative Earth’s magnetic field (EMF) generator in conjunction with SGP-4-based satellite orbital propagator high-level control software. Additionally, cascade proportional-integral-derivative (PID) and state-dependent Riccati equation (SDRE) controllers are implemented to generate sufficient torque using three-axis magnetorquers on a frictionless air-bearing platform. The platform is balanced to closely simulate the dynamic motion of a spacecraft in space. The testing includes a single initial condition and three inertia conditions for stabilization after solar panel wing deployment. Finally, the effectiveness of the cosimulation as a primary experiment through an integrated HiLs process is validated. This comprehensive approach confirms the control system’s performance and its ability to meet mission requirements.

Dynamics analysis and chatter control of a polishing and milling nonideal flexible manipulator

Some robots are designed to be lightweight and flexible, enabling them to access small and challenging paths in various applications. These features enable robots to collaborate with humans in performing specific production tasks. However,the movement of the flexible manipulator can become self-excited when handling a cutting tool on a workpiece, which can lead to a control problem. This article presents a control solution for lightweight robotic manipulators with rotating tools, such as polishing and milling, using smart actuators. The control discretization method is also introduced to facilitate integration into digital controllers. The paper starts by describing the governing equations of the non-ideal flexible manipulator for polishing and milling and analysing its dynamic behavior. Subsequently, models for the controllers of the DC motor-only actuators and the hybrid (shape memory alloy and DC motors) actuators were formulated using shape memory alloy and a suboptimal control scheme known as the discrete state-dependent Riccati equation. The coupled system was analysed dynamically, and it was observed that it exhibits chaotic behavior. The results suggest that incorporating smart actuators into the motor group of the system could decrease the positioning error of the manipulator and significantly reduce the oscillation of the robot’s end-effector.

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.

Cobra aerobatic flight for quadrotors using θ-D nonlinear control

The control design for the Cobra maneuver of quadrotors is presented based on quaternion dynamics. Cobra maneuver commands the quadrotor in ± π / 2 (rad) pitch angle such that the conventional hovering assumption for quadrotor flight is not applicable. The finite-time θ-D method is proposed to control the Cobra maneuver. Based on this technique, the intractable Hamilton–Jacobi–Bellman equation can be approximated by a differential Riccati equation and a series of state-dependent Lyapunov equations, which can be solved recursively and lead to the closed-form state feedback control law . This control law can achieve semi-global asymptomatic stability of the closed-loop system. Numerical simulations verify that the proposed technique is compared favourably with the finite-time state-dependent Riccati equation (SDRE) technique in terms of computation efficiency and control effort, and also shows faster recovery of the quadrotor after the Cobra maneuver compared with conventional dynamics without the derivative of the rotation matrix.

Design an omnidirectional autonomous mobile robot based on non‐linear optimal control to track a specified path

AbstractThis paper explores two non‐linear control techniques for designing an effective control system for an omnidirectional autonomous mobile robot with four Mecanum wheels. Due to the unique wheel structure and four separate wheels, the robot has non‐linear dynamics, multiple inputs and outputs. The first technique uses the state‐dependent Riccati equation (SDRE) to address optimal non‐linear control while considering energy and time constraints. The second technique, using an intermediate variable , has expanded the Hamilton‐Jacobi‐Belman equation in terms of the power series. Consequently, these equations are reduced to a set of recursive Lyapunov algebraic equations, leading to a closed‐form solution for solving the non‐linear optimal control problem. Finally, the maneuverability and path‐tracking capability of the robot are examined by highlighting the non‐linear term through numerical simulation.

Energy-optimal adaptive artificial potential field method for path planning of free-flying space robots

Aiming at planning a collision-free path for a free-flying space robot (FFSR), an energy-optimal collision avoidance strategy is proposed using the adaptive artificial potential field (AAPF) approach. To address the goals non-reachable with obstacles nearby (GNRON) issues of artificial potential field (APF), an adaptive and secure path planning method is proposed by incorporating a smooth decay function into the repulsive potential field. Meanwhile, a time-varying shaping parameter of attractive potential field is optimized combined with the state-dependent Riccati equation technique to reduce the energy cost of the FFSR. In case the system gets trapped in local minimum, an escape force is elaborately designed to replace artificial potential force based on optimal theory, whose direction vector guides the FFSR to jump out of the local minimum with minimal energy. Finally, simulation results on a planar FFSR demonstrate the effectiveness of the proposed methodology and show improved performance compared with the existing artificial potential field (APF) in terms of escaping local minimum and reducing energy consumption.

Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

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.

A suboptimal control scheme based on integrated real-time nonlinear estimation-identification approach for developed flexible joint manipulators structure

As the safety of human-robot interaction is of great interest in various applications, the flexibility of machine joints becomes increasingly important. In this paper, a general model is developed for electrically driven flexible-joint robots (ED-FJR), in which the coupled phenomena of stiffness-damping and friction in the joints are modeled as nonlinear and time-varying parameters. Because the output torque of the actuator mounted on the link cannot be measured, its state variables are estimated by the suboptimal estimator. Furthermore, the unknown parameters of joint flexibility are identified by the identification algorithm based on time-varying nonlinear least-squares error (TV-NLS), which are then used in the proposed control algorithm to compensate for negative effects and errors. The state-dependent Riccati equation (SDRE) method is developed in this paper using the combined estimator-identifier approach. The novel proposed method for a 3-degree-of-freedom (3-DOF) mechanical arm with electrically driven flexible joints is implemented and validated. The obtained results show that the system's performance has improved by 83% in terms of modeling accuracy. Furthermore, when compared to other methods, the proposed algorithm reduces tracking error by 59%, making it easier to reach the target point.

A robust state-dependent Riccati equation controller with parameter uncertainty and matched disturbance

The state-dependent Riccati equation (SDRE) unveils a nonlinear optimal control approach; the method is sensitive to uncertainty and disturbance that provokes the necessity of precise modeling and omitting disturbance from the model and environment which is occasionally impossible. In this paper, a robust state-dependent Riccati equation is introduced using only the pure SDRE itself. Lyapunov’s second method for stability analysis is used to extract two tuning rules for weighting matrices of the robust approach for handling both matched disturbance and parametric uncertainty. A fourth-order differential equation is simulated to show the effectiveness of the suggested method and the results were compared with conventional SDRE. The robust version could handle the matched disturbance with a steady-state shift from the equilibrium point where the conventional SDRE failed. Additionally, a comparison and detailed analysis have been performed between the proposed design and sliding mode control.

Guidance strategy of motion camouflage for spacecraft pursuit-evasion game

This work is inspired by a stealth pursuit behavior called motion camouflage whereby a pursuer approaches an evader while the pursuer camouflages itself against a predetermined background. We formulate the spacecraft pursuit-evasion problem as a stealth pursuit strategy of motion camouflage, in which the pursuer tries to minimize a motion camouflage index defined in this paper. The Euler-Hill reference frame whose origin is set on the circular reference orbit is used to describe the dynamics. Based on the rule of motion camouflage, a guidance strategy in open-loop form to achieve motion camouflage index is derived in which the pursuer lies on the camouflage constraint line connecting the central spacecraft and evader. In order to dispose of the dependence on the evader acceleration in the open-loop guidance strategy, we further consider the motion camouflage pursuit problem within an infinite-horizon nonlinear quadratic differential game. The saddle point solution to the game is derived by using the state-dependent Riccati equation method, and the resulting closed-loop guidance strategy is effective in achieving motion camouflage. Simulations are performed to demonstrate the capabilities of the proposed guidance strategies for the pursuit–evasion game scenario.

Nonlinear optimal control for permanent magnet synchronous spherical motors

Nonlinear optimal control for permanent magnet synchronous spherical motors

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.

Finite-Time Suboptimal Control Design for Aerobatic Maneuver of Variable-Pitch Quadrotor

The variable-pitch quadrotors can perform aggressive maneuvers such as aerobatic flip by varying the thrust in both positive and negative vertical directions via the blade pitch, which can provide an additional degree of freedom as opposed to the fixed-pitch quadrotor that only changes the rotational rate of motors. However, an inevitable challenge exists when designing an operative controller for the aerobatic flip due to the resultant singularity of the rotation matrix. In this paper, a finite-time suboptimal control design strategy named finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> algorithm combined with the geometric control technique is proposed for the aerobatic maneuver control of the variable-pitch quadrotor. Based on this scheme, the singularity issue can be circumvented. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> control design provides a closed-form suboptimal control law by offline solving a differential Riccati equation and a set of linear Lyapunov equations analytically, which does not require complex online numerical and iterative solutions compared with the similar finite-time state dependent Riccati equation technique. Simulation results demonstrate that the proposed geometric finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> method provides an effective and efficient control of the variable-pitch quadrotor.

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.

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.

Statistical Investigation about the SDRE Optimality for Satellites with Nonlinearities

The Attitude Control System (ACS) of a Satellite can be designed with success by linear control theory if the satellite has slow angular motions. However, for fast maneuvers, the linearized models are not able to represent the effects of the nonlinear terms. One candidate technique to design the ACS of the satellite under fast maneuvers is the State-Dependent Riccati Equation (SDRE), which provides an effective algorithm for synthesizing nonlinear feedback control taking into account the nonlinearities. Nonetheless, much criticism has been leveled against the SDRE method since it does not assure global asymptotic stability (GAS), because there are situations in which global asymptotic stability cannot be achieved (e.g., systems with multiple equilibrium points). One way to study the GAS is by estimating the region of attraction (ROA) which in turn is fundamental to investigate the performance of the controller designed by the SDRE method The Brazilian National Institute for Space Research (INPE) is responsible to build a Remote-Sensing Satellites, called Amazonia-1. The ACS of the Amazonia-1 satellite must be stabilized in three axes so that the optical payload can point to the desired target. In this paper, one investigates the performance of the Amazonia-1 ACS using a statistical test (unpaired t-test) that compares the optimality inside the ROA controlled by LQR and SDRE. By several simulations running a full Monte Carlo perturbation satellite model one observes a significant difference between the optimality of the two controllers in favor of SDRE when nonlinearities are accounted for.

Finite-time integral fast terminal sliding mode control for uncertain quadrotor UAV based on state-dependent Riccati equation observer subjected to disturbances

This paper presents a new control approach of trajectory tracking for quadrotor unmanned aerial vehicle (UAV) systems under external disturbances and parameter uncertainties. The integral fast terminal sliding mode control (IFTSMC) approach applying state-dependent Riccati equation (SDRE) observer is presented for this purpose. As a result, the nonlinear SDRE observer is designed to estimate the non-measurable states. Afterward, employing the estimated states of the SDRE observer, the proposed IFTSMC controller is designed for trajectory tracking of the perturbed quadrotor UAV. Besides, applying Lyapunov’s stability theory, the finite-time convergence and robustness of the closed-loop UAV system are assured. Based on the numerical simulation results, the proposed controller successfully tracked various reference trajectories of quadrotor UAVs despite external disturbances and parametric uncertainty.

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

Directed Distance-Based Formation Control of Nonlinear Heterogeneous Agents in 3-D Space

This paper studies distance-based formation control of a set of nonlinear multi-agent systems over directed graphs. We propose a distributed, distance-based formation control scheme for a set of heterogeneous, nonlinear agents over a particular class of minimally, structurally persistent, directed graphs in a 3-D space, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">directed trilateral Laman</i> graphs. The responsibility of controlling each directed edge is assigned to only one of the adjacent agents. The state-dependent Riccati equation is used to design the control method for nonlinear agents. Based on the mathematical induction and stability theory of cascade interconnected systems, we rigorously prove the asymptotic stability of the overall formation. A combination of signed area and volume constraints is used to prevent agents from converging to the flip-ambiguous frameworks in 3-D space. The proposed control law assures collision avoidance between the neighboring pairs of agents. Simulation results are provided to verify the theoretical results.