State-dependent Riccati Equation Research Articles

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

PurposePermanent magnet synchronous spherical motors can have wide use in robotics and industrial automation. They enable three-DOF omnidirectional motion of their rotor. They are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. Unlike conventional synchronous motors, permanent magnet synchronous spherical motors consist of a fixed inner shell, which is the stator, and a rotating outer shell, which is the rotor. Their dynamic model is multivariable and strongly nonlinear. The treatment of the associated control problem is important.Design/methodology/approachIn this paper, the multivariable dynamic model of permanent magnet synchronous spherical motors is analysed, and a nonlinear optimal (H-infinity) control method is developed for it. Differential flatness properties are proven for the spherical motors’ state-space model. Next, the motors’ state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point, which is defined by the present value of the motors’ state vector and by the last sampled value of the control input vector. For the approximately linearized model of the permanent magnet synchronous spherical motors, a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains, an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops.FindingsDue to the nonlinear and multivariable structure of the state-space model of spherical motors, the solution of the associated nonlinear control problem is a nontrivial task. In this paper, a novel nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of permanent magnet synchronous spherical motors. The method is based on approximate linearization of the motor’s state-space model with the use of first-order Taylor series expansion and the computation of the associated Jacobian matrices. Furthermore, the paper has introduced a different solution to the nonlinear control problem of the permanent magnet synchronous spherical motor, which is based on flatness-based control implemented in successive loops.Research limitations/implicationsThe presented control approaches do not exhibit any limitations, but on the contrary, they have specific advantages. In comparison to global linearization-based control schemes (such as Lie-algebra-based control), they do not make use of complicated changes of state variables (diffeomorphisms) and transformations of the system's state-space description. The computed control inputs are applied directly to the initial nonlinear state-space model of the permanent magnet spherical motor without the intervention of inverse transformations and thus without coming against the risk of singularities.Practical implicationsThe motion control problem of spherical motors is nontrivial because of the complicated nonlinear and multivariable dynamics of these electric machines. So far, there have been several attempts to apply nonlinear feedback control to permanent magnet-synchronous spherical motors. However, due to the model’s complexity, few results exist about the associated nonlinear optimal control problem. The proposed nonlinear control methods for permanent magnet synchronous spherical motors make more efficient, precise and reliable the use of such motors in robotics, electric traction and several automation systems.Social implicationsThe treated research topic is central for robotic and industrial automation. Permanent magnet synchronous spherical motors are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. The solution of the control problem for the nonlinear dynamic model of permanent magnet synchronous spherical motors has many industrial applications and therefore contributes to economic growth and development.Originality/valueThe proposed nonlinear optimal control method is novel compared to past attempts to solve the optimal control problem for nonlinear dynamical systems. Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation which is used for computing the feedback gains of the controller is new, and so is the global stability proof for this control method. Compared to nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed into the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes, which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions. Furthermore, the second control method proposed in this paper, which is flatness-based control in successive loops, is also novel and demonstrates substantial contribution to nonlinear control for robotics and industrial automation.

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

PurposeVoltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSI-PMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSI-PMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSI-fed three-phase PMSMs.Design/methodology/approachThe present article proposes a nonlinear optimal control approach for VSI-PMSMs. The nonlinear dynamic model of VSI-PMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is defined by the present value of the voltage source inverter-fed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverter-fed PMSM, an H-infinity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSI-fed PMSM under moderate variations of the control inputs.FindingsThe proposed H-infinity controller provides the solution to the optimal control problem for the VSI-PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min–max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI-PMSM loop has the H-infinity tracking performance property, which signifies robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSI-PMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSI-PMSM control loop can be minimized.Practical implicationsThe proposed nonlinear optimal control method for the VSI-PMSM system exhibits several advantages: Comparing to global linearization-based control methods, such as Lie algebra-based control or differential flatness theory-based control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSI-PMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state-space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with sliding-mode control, there is no need to define in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local model-based control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort.Social implicationsVoltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSI-fd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSI-fed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective.Originality/valueUnlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.

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.

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.

Formation flying along artificial halo orbit around Sun–Earth L2 point for interferometric observations

This paper proposes a method of using multiple spacecraft flying in formation in an artificially reduced halo orbit around the 2nd Lagrange point in the Sun–Earth system (SEL2) to make interferometric observations. One of the requirements for interferometric observations is to collect baseline vectors between telescopes or antennas. A shape-based approach is introduced to design a formation flight orbit that spirally spreads to satisfy this requirement. In this approach, orbits are designed on the basis of a linear theory. However, the formation orbits based on the linear theory quickly diverge because of the unstable dynamics around the SEL2. Therefore, the state-dependent Riccati equation, which allows for nonlinear control, is applied to orbit maintenance. The control system consists of two components: one for maintaining the reference orbit around the SEL2 and the other for controlling the relative position between the multiple spacecraft. The performance of the developed formation flight system is verified through a numerical simulation, confirming that the position accuracy requirement of the infrared interferometer is satisfied. It is also shown that both controls can be achieved with a low-thrust magnitude by using an electric propulsion system.

State-Dependent Riccati Equation-Based UAV Path Following Guidance for Shipborne Net Recovery

The landing of a fixed-wing aircraft on a ship is a challenging task due to a constrained landing space, wind disturbances, and the ship motion. In this paper, we consider the landing problem as a moving path following problem and present a decoupled three-dimensional State-Dependent Riccati Equation (SDRE)-based guidance law for net recovery landing of a fixed-wing unmanned aerial vehicle (UAV) on board a ship deck. Stability analysis of the guidance law is carried out and its performance is evaluated on aerosonde UAV in MATLAB and RASCAL aircraft in software-in-the-loop (SITL) taking actuator dynamics, different sea states for ship motion and varying wind disturbances into account. The proposed approach shows the general applicability of the guidance law for different types of aircraft