Derivation Of Control Law Research Articles

A lie group PMP approach for optimal stabilization and tracking control of autonomous underwater vehicles

AbstractIn this research, we explore a finite horizon optimal stabilization and tracking control scheme for the dynamical model of a 6‐DOF Autonomous Underwater Vehicle (AUV). Dynamical equations of the AUV are represented in a Lie group () framework, encompassing both translational and rotational motions. Utilizing a left Lie group action on , we define error function for velocities via a right transport map to effectively address optimal trajectory tracking. The optimal control objective is formulated as a trade‐off problem, aiming to minimize both errors and control effort simultaneously. Left action on yields the left trivialized Hamiltonian function from which the concomitant state and costate dynamical equations are derived using Pontryagin's Minimum Principle (PMP). Consequently, the resulting two‐point boundary value problem is solved to obtain optimal trajectories. We demonstrate the optimality of the resulting solution obtained from the derived control law. For ensuring boundedness in the presence of small disturbances, this study incorporates the effects of internal parametric uncertainties associated with added mass and inertia components, along with the influence of external disturbances induced by ocean currents. Through simulation validations, we confirm the alignment of our results with the theoretical developments, demonstrating that the proposed control law effectively mitigates both parametric uncertainties and ocean current disturbances.

Feedback boundary control of multi-dimensional hyperbolic systems with relaxation

This paper is concerned with boundary stabilization of multi-dimensional hyperbolic systems of partial differential equations. By adapting the Lyapunov function previously proposed by the second author for linearized hyperbolic systems with relaxation structure, we derive certain control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to water flows in open channels. The effectiveness of the derived control laws is confirmed by numerical simulations.

Asymptotic tracking control for a class of uncertain nonlinear systems with output constraint

In this paper, we investigate the asymptotic tracking control for a class of strict-feedback nonlinear systems with unknown nonlinear functions and output constraint. The nonlinear transformation function-based system transformation approach is employed to convert the initial constrained system into an unconstrained one. To achieve the objective of asymptotic tracking, the tracking error is scaled by an exponential proportional function. Furthermore, to address the challenge of repetitive derivation of virtual control laws that occurs in the traditional design process of high-order backstepping methods, a first-order filter is applied. The designed control scheme guarantees that the system output will asymptotically track a predefined reference tracking trajectory while ensuring that the output constraint is not violated. Moreover, it is shown that the filter error will converge asymptotically to the origin. The effectiveness of the developed control strategy is verified through simulation results.

Fixed-time consensus control for uncertain heterogeneous multi-agent systems with high-order dynamics and time-varying delay under generic topologies

This article is concerned with the fixed-time control problem of uncertain high-order heterogeneous multi-agent systems with time-varying delays for both directed and undirected topologies. Applying the free-weighting matrix approach, a sliding mode controller is designed through the Lyapunov–Krasovskii functional and the conditions are derived with the help of linear matrix inequalities to attain fixed-time consensus tracking control of such systems. Two numerical simulations are included to verify the capability of the derived control law with respect to different initializations.

Automatic design of interpretable control laws through parametrized Genetic Programming with adjoint state method gradient evaluation

This work investigates the application of a Local Search (LS) enhanced Genetic Programming (GP) algorithm to the control scheme’s design task. The combination of LS and GP aims to produce an interpretable control law as similar as possible to the optimal control scheme reference. Inclusive Genetic Programming (IGP), a GP heuristic capable of promoting and maintaining the population diversity, is chosen as the GP algorithm since it proved successful on the considered task. IGP is enhanced with the Operators Gradient Descent (OPGD) approach, which consists of embedding learnable parameters into the GP individuals. These parameters are optimized during and after the evolutionary process. Moreover, the OPGD approach is combined with the adjoint state method to evaluate the gradient of the objective function. The original OPGD was formulated by relying on the backpropagation technique for the gradient’s evaluation, which is impractical in an optimization problem involving a dynamical system because of scalability and numerical errors. On the other hand, the adjoint method allows for overcoming this issue. Two experiments are formulated to test the proposed approach, named Operator Gradient Descent - Inclusive Genetic Programming (OPGD-IGP): the design of a Proportional–Derivative (PD) control law for a harmonic oscillator and the design of a Linear Quadratic Regulator (LQR) control law for an inverted pendulum on a cart. OPGD-IGP proved successful in both experiments, being capable of autonomously designing an interpretable control law similar to the optimal ones, both in terms of shape and control gains.

An improved design for virtual synchronous generator control loop based on synergy theory

Virtual synchronous generator (VSG) technology has achieved some results in distributed generation networks and enhanced system stability. For problems that may occur in the system due to ignoring parameter linkages, this paper designs a synergistic controller that establishes the connection between the active frequency and the reactive voltage control loop based on synergy theory. As the output of the synergetic controller, the derived control law u is added to the power frequency control in the form of negative feedback. Two disturbance forms, a step disturbance, and a three-phase short-circuit fault, are set up. The conventional droop control and synergetic droop control results are compared using a MATLAB simulation system. Then, the system parameter variations were studied and analyzed, and the sensitivity of the control system with different values of parameters K1, K2, and K3 in the case of a three-phase short circuit was analyzed, which provides a reference guide for selecting parameters in later control optimization. Finally, the control effectiveness of a multi-machine VSG system was tested. The simulation results show that the proposed method has a pronounced effect on making the system stabilize quickly, which illustrates the effectiveness of this technique.

A Parallel Control Method For Zero-Sum Games With Unknown Time-varying System

In this paper, based on ACP approach, a parallel control method is proposed for zero-sum games of unknown time-varying systems. The process of constructing a sequence of artificial systems, implementing the computational experiments and conducting the parallel execution is presented. The artificial systems are constructed to model the real system. Computational experiments adopting adaptive dynamic programming (ADP) are shown to derive control laws for the a sequence of artificial systems. The purpose of the parallel execution step is to derive the control laws for the real system. Finally, simulation experiments are provided to show the effectiveness of the proposed method.

Parameter Identification for Maximum Torque per Ampere Control of Permanent Magnet Synchronous Machines under Magnetic Saturation

This paper applies the identified parameters of permanent magnet synchronous machines (PMSMs) for the maximum torque per ampere control (MTPA) under magnetic saturation. The variation in magnet flux with current is determined using a position offset approach while the variation in q-axis inductance with the current is estimated from the d-axis voltage equation afterward. In addition, the d-axis inductance is estimated at standstill by the injection of a small amplitude of high frequency d-axis current. The curve-fitted results of estimated parameters under different saturation conditions are then employed to aid the derivation of MTPA control law. The proposed method is experimentally verified on two prototype PMSMs. Experimental results show that compared with conventional MTPA schemes using fixed values of magnetic parameters, the proposed method can increase maximum output torque by 2.1% and 3.2% on two prototype PMSMs, respectively.

Utilizing a control technique for orbital maintenance near [formula omitted] point and Lyapunov exponents

In present article, we have explored the orbit maintenance strategy around an artificial equilibrium point in the Sun–Jupiter system. Two control strategies, proportional control and proportional derivative control laws are discussed for orbit maintenance. The maximum Lyapunov exponent is used to check the chaotic behavior of planetary orbit. We anticipate that the recently established Lyapunov exponent criteria may be used to study motion of planets in generalized stellar binary systems. This research is being guided by the conclusions of recent observations as well as those that are expected from existing and upcoming planet search missions.

Optimal tracking control of an Autonomous Underwater Vehicle: A PMP approach

This paper addresses the optimal tracking control problem of an Autonomous Underwater Vehicle (AUV) using Pontryagin’s Minimum Principle (PMP). The formulation uses a choice of Hamiltonian function using the desired control objective with AUV dynamics acting as dynamic constraints. We first develop PMP based on AUV kinematics and then extend it to the dynamics to arrive at optimal thrusts and moments. Necessary conditions for optimality are derived for both the models using PMP, which results in optimal trajectories that simultaneously minimize the tracking error as well as the control cost, thereby arriving at energy optimality. It was observed that the adjoint variables (costates) are indeed the momenta in the inertial and body-fixed frames. At the kinematic level, this forms a stable solution. The developed methodology is applied to both 2D and 3D AUV model with disturbances due to inputs and ocean currents. Numerical simulations are carried out with the derived control laws for a given trajectory tracking target. Quantitative evaluation of the performance and comparison of the controller is done using Mean Square Error(MSE) and Total Variation(TV) measures. The proposed control laws are found to achieve the desired control objectives.

Minimization of acoustic power in free space using dipole sound sources

Low-frequency noise is difficult to suppress by passive control using a sound absorbing material. Active noise control is expected to be an effective method against low-frequency noise. Minimizing the total acoustic power emitted by a sound source causes a zero-control power phenomenon. In the zero control power phenomenon, the absorption and blowout of sound energy around the control sound source (CSS) are balanced, and the time average of the control power becomes zero. Therefore, control power suppression may be able to minimize total acoustic power. Previous studies have assumed that the sound source is a monopole sound source or a piston embedded in an infinite baffle. In this research, the sound sources are regarded as dipole sound source. In addition, numerical calculations are performed based on the derived control law to clarify the effect of the position and the rotation angle of the dipole on the acoustic power.

Repetitive process based indirect-type iterative learning control for batch processes with model uncertainty and input delay

This paper develops an indirect iterative learning control scheme for batch processes with time-varying uncertainties, input delay, and disturbances. In this paper, a predictor based on a state observer is designed to estimate the future state and to compensate for the input delay. Then a feedback controller based on the estimated state and the set-point error is used to track the specified reference trajectory, where, of the options available, a robust H∞ controller is designed in the presence of time-varying uncertainties and load disturbances. Then a proportional plus derivative type iterative learning control law is designed. An injection molding process model demonstrates the new method’s effectiveness, and a comparison with a direct-type design is given.

A time domain approach for the optimal control of wave energy converter arrays

Wave energy converters typically use various control methods to extract energy from ocean waves. The objective of the control system is to optimize the energy extraction process, taking into account the dynamics of the system and the wave conditions. The task of deriving the optimal control laws of wave energy converter arrays for regular and irregular waves using the Pontryagin minimum principle was previously investigated in the literature. The result is a combination between the singular arc and bang-bang control laws. For irregular waves, some complexity arises due to the radiation state-space representation, which requires ignoring the hydrodynamic coupling terms related to the added mass and radiation-damping coefficients; this reduces the computational complexity of the control force but adversely affects the solution's accuracy. Also, the derived control laws are specific to a particular wave condition. Recently, the optimal control of a flexible buoy wave energy converter was derived using the convolution representation for the radiation force. In this work, the optimal control laws of flexible buoy wave energy converters are modified to simulate wave energy converter arrays; then, the results are compared to those obtained by dropping the hydrodynamic radiation coupling terms. Although using a convolution representation adds computational complexity to the optimal control problem, it generates an equation that is generic to any wave condition, can be used with any wave spectrum, and provides an expression for the switching condition.

Application of Impedance Control of the Free Floating Space Manipulator for Removal of Space Debris

A broad and significant class of space debris can be mitigated by means of a satellite, capable of capturing a large non-cooperating object by using a robotized arm with a gripper. The capture operation typically comprises of an approach, a close-on manoeuvre, establishing contact between the robotic grappler arm and a suitable feature on the target satellite, and finally it is concluded when a positive mechanical connection is achieved by the gripper closed on that feature. The phase of establishing contact poses a critical challenge in this scenario, since the target typically will be tumbling with respect to the chaser satellite causing high forces on the gripper and the robotic arm. A family of control methods known collectively as impedance control is typically employed in terrestrial robots for tasks involving an interaction with an environment, especially the dynamic contact. In this work, we present the model-based impedance control applied to a robotic manipulator on a free floating base. The derivation of impedance control law for a robotic manipulator on a free floating satellite, involving Generalized Jacobian Matrix (GJM), is presented, followed by simulation results comparing the loads in the manipulator joints against a classical GJM-based Cartesian controller. The simulation results show that the impedance controlled free floating robotic manipulator completes the task of trajectory following amid contact with unknown target with lower torques in the robot joints.

Variations in Finite-Time Multi-Surface Sliding Mode Control for Multirotor Unmanned Aerial Vehicle Payload Delivery with Pendulum Swinging Effects

Multi-surface sliding mode control addresses the limitations of traditional sliding mode control by employing multiple sliding surfaces to handle uncertainties, disturbances, and nonlinearities. The design process involves developing sliding surfaces, designing switching logic, and deriving control laws for each surface. In this paper, first, a robust finite-time multi-surface sliding mode controller will be presented and its performance analyzed by applying it to a multirotor subjected to a suspended payload, modeled in the form of a single pendulum, itself defined as a spatial (3D) dynamic model. Next, an adaptive finite-time multi-surface sliding mode controller will be derived—adding a variable adaptive parameter to the existing sliding surfaces of the robust finite-time control—and applied to the same system. It will be shown that the adaptive controller, with an adaptive parameter that adjusts itself based on the present value of the multi-surface sliding mode parameter, creates an improved fast finite-time convergence by obtaining an optimal settling time and minimizing undershoot of the multirotor state vector. Empirical verification of the effectiveness of the adaptive control will be carried out by presenting the control performances against a step response. It is also shown that the control may be utilized to approximate external disturbances—represented by the pendulum—and that with the application of control, the vehicle’s motion may be stabilized and the payload swing suppressed. Lyapunov stability theory-based stability proofs for the controllers’ designs are developed, showing the asymptotic stability of the output and uniform boundedness of the errors in the system dynamics. It is verified that the multi-surface sliding mode control can account for system uncertainties—both matched and mismatched—in addition to changes in internal dynamics and disturbances to the system, where the single pendulum payload is representative of the changes in dynamics that may occur to the system. Numerical simulations and characteristics are presented to validate the performance of the controllers.

A spatiotemporal fuzzy control approach for complex distributed parameter system with application to thermal processes

Many industrial processes belong to distributed parameter systems (DPSs) with unknown spatiotemporal dynamics and complex boundary conditions. However, attentions to both temporal and spatial dynamics in the control law are less paid in the existing control methods, which often makes them challenging to achieve accurate control. Here, a spatiotemporal fuzzy control approach with consideration of both temporal and spatial dynamics is proposed for unknown and complex DPSs. First, a spatiotemporal fuzzy model is developed to reconstruct dynamics of nonlinear DPS. Since it constructs a two-layer fuzzy rules (i.e., temporal fuzzy rule and spatial fuzzy rule) to consider both the nonlinear dynamics on time and the nonlinear energy transfer on space, it can effectively achieve the model of DPSs. On this basis, a spatiotemporal fuzzy controller is developed according to the developed two-layer fuzzy rules, which means that this derived control law considers both temporal and spatial dynamics. Then, a two-step solution method is designed for the spatiotemporal control law. Stability conditions of the control system are rigorously deduced by transforming the solving process to a series of linear matrix inequalities (LMIs). The stability and effect of the proposed spatiotemporal control strategy are demonstrated by mathematical proof and case studies. Comparison results demonstrate that the proposed method has better control performance (relative errors and mean absolute percentage errors are bounded in 4%).

Adaptive Fixed-Time Tracking Control for Nonlinear Systems Based on Finite-Time Command-Filtered Backstepping

The paper takes into account the problem of adaptive fixed-time control for nonlinear systems in strict form via finite-time command filtered backstepping. Our presented control scheme gives consideration to rapidity by using fixed-time control and finite-time command filter, and its prime control objective is to ensure that the system output can be guided from any initial conditions to go after an ideal variable. Meanwhile, this control strategy makes sure that all the system states and other signals are bounded at finite time, and its convergence time doesn't have no connection with the system initial conditions and is determined by the design parameters. At every step in the design process, a fuzzy logic system is introduced to approximate the unknown part, and a finite-time command filter is also utilized so as to avoid the issue of derivation of virtual control laws and complex calculation. At last, we verify that the presented scheme is viable through a simulation example.

Robust steering control for a steer-by-wire automated guided vehicle via fixed-time adaptive recursive sliding mode

In this paper, a fixed-time adaptive recursive sliding mode (FTARSM) control scheme is addressed for a steer-by-wire (SbW) automated guided vehicle against model uncertainties and disturbances. First, based on a newly constructed faster fixed-time stable system, a fixed-time recursive sliding structure is developed to guarantee the SbW system fixed-time convergence, where the setting time is independent of initial conditions. By making appropriate initialization settings for the recursive structure, the sliding reaching phase is removed and the control robustness is improved. Then, the extreme learning machine (ELM) is incorporated into the FTARSM controller to estimate the lumped uncertainties upper bound, thus not only the requirement for prior bounds information in controller design is eliminated but also the control chattering is suppressed effectively. Rigorous Lyapunov analyses are further employed to ensure fixed-time closed-loop stability. Finally, the superior performance of the derived control law is verified by experimental results.

Adaptive Sliding Mode Control with Chattering Elimination for Buck Converter Driven DC Motor

The Adaptive Sliding Mode Control (ASMC) that combines a robust proportional derivative control law for use in Buck converter driven DC motor is presented in this paper. Based on the LYAPUNOV theory, the proportional derivative control law is designed to eliminate the chattering action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability in the LYAPUNOV sense if all the signals involved are uniformly bounded. Simulation studies have shown that the proposed controller shows superior tracking performance.

Integral Backstepping Control Algorithm for a Quadrotor Positioning Flight Task: A Design Issue Discussion

For quadrotor control applications, it is necessary to rely on attitude angle changes to indirectly achieve the position trajectory tracking purpose. Several existing literature studies omit the non-negligible attitude transients in the position controller design for this kind of cascade system. The result leads to the position tracking performance not being as good as expected. In fact, the transient behavior of the attitude tracking response cannot be ignored. Therefore, the closed-loop stability of the attitude loop as well as the position tracking should be considered simultaneously. In this study, the flight controller design of the position and attitude control loops is presented based on an integral backstepping control algorithm. This control algorithm relies on the derivatives of the associated virtual control laws for implementation. Examining existing literature, the derivatives of the virtual control law are realized approximated by numerical differentiations. Nevertheless, in practical scenarios, the numerical differentiations will cause the chattering phenomenon of control signals in the presence of unavoidable measurement noise. The noise-induced control signals may further cause damage to the actuators or even diverge the system response. To address this issue, the analytic form for the derivative of the virtual control law is derived. The time derivative virtual control law is analyzed and split into the disturbance-independent compensable and disturbance-dependent non-compensable terms. By utilizing the compensable term, the control chattering due to the differentiation of the noise can be avoided significantly. The simulation results reveal that the proposed control algorithm has a better position tracking performance than the traditional dual-loop control scheme. Meanwhile, a relatively smooth control signal can be obtained for a realistic control algorithm realization. Simulations are provided to illustrate the position tracking issue of a quadrotor and to demonstrate the effectiveness of the proposed compromised control scheme.