Lyapunov-based Control Law Research Articles

Solar sailing adaptive control using integral concurrent learning for solar flux estimation

In the interest of exploiting natural forces for propellant-less spacecraft missions, this investigation proposes an adaptive control strategy to account for unknown parameters in the dynamic modeling of a reflectivity-controlled solar sail spacecraft. A Lyapunov-based control law along with integral concurrent learning is suggested to accomplish and prove global exponential tracking of the estimated parameters and states of interest, without satisfying the common persistence of excitation condition, which in most nonlinear systems cannot be guaranteed a priori. This involves estimating the solar flux or irradiance from the Sun to account for uncertainty and variation over time in this value. To illustrate potential applications, two missions are considered: (1) a geostationary debris removal case and (2) an Earth–Mars interplanetary transfer orbit following a logarithmic spiral reference trajectory. The proposed formulation demonstrates the benefit of estimating the solar flux using integral concurrent learning. The results are compared to trajectories with no estimation to illustrate the need to account for the actual solar flux.

Overcoming backstepping limitations via a novel MIMO non-affine-in-control convex optimization technique

Limitations of the standard backstepping control technique are overcome in this paper by yielding global conditions for regional ones; specifically, flexible cascade arrangements that do not require the system to be put in a pure- or strict-feedback form to ensure controllability beforehand, handling of nonlinear MIMO blocks without invertible input distribution matrices, calculation of virtual and real control laws regardless of the non-affine-in-control nature of subsystems, and systematic numerical synthesis of Lyapunov-based control laws at each step of the cascade connection via linear matrix inequalities. All these benefits derive from exploiting exact convex rewriting of nonlinear terms, splitting available and non-available signals for control implementation, and applying the direct Lyapunov method to cast conditions as a convex optimization problem. Examples of academic and practical interest illustrate the advantages of the novel methodology over former approaches.

Preparation of the Hadamard and CNOT gates to realize the maximum entangled states

This paper studies the preparation of quantum gates that enable the realization of maximum entangled states, i.e., Bell states. A series of unitary time-evolution operators are derived to realize the Bell states through the Hadamard and CNOT gates. These operators are decomposed based on the Cartan decomposition method and steered by control laws designed based on the Lyapunov control theorem. The numerical simulations show that by adjusting proper control parameters in the designed Lyapunov-based control laws, the Hadamard and CNOT gates can be faster and better prepared through the decomposed operators, and more stable and accelerated Bell states can be achieved.

Optimal Q-laws via reinforcement learning with guaranteed stability

Closed-loop feedback-driven control laws can be used to solve low-thrust many-revolution trajectory design and guidance problems with minimal computational cost. Lyapunov-based control laws offer the benefits of increased stability whilst their optimality can be increased by tuning their parameters. In this paper, a reinforcement learning framework is used to make the parameters of the Lyapunov-based Q-law state-dependent, increasing its optimality. The Jacobian of these state-dependent parameters is available analytically and, unlike in other optimisation approaches, can be used to enforce stability throughout the transfer. The results focus on GTO–GEO and LEO–GEO transfers in Keplerian dynamics, including the effects of eclipses. The impact of the network architecture on the behaviour is investigated for both time- and mass-optimal transfers. Robustness to navigation errors and thruster misalignment is demonstrated using Monte Carlo analyses. The resulting approach offers potential for on-board autonomous transfers and orbit reconfiguration.

Lyapunov-based economic model predictive control for nonlinear descriptor systems

A large number of physical applications such as electrical circuits, chemical processes and multibody systems can be accurately described by a set of differential algebraic equations (DAEs). For instance, sustainable water desalination techniques such as membrane distillation (MD) is an example of a chemical process that is described by DAEs. Optimal control strategies such as economic model predictive control (EMPC) has many operational advantages (e.g., maximizing process economics and enhancing operational safety) for chemical applications. However, constructing EMPC schemes for a general class of nonlinear DAEs systems while ensuring closed-loop stability and recursive feasibility is an open research challenge that has not been considered. Motivated by the above considerations, this note introduces a Lyapunov-based economic model predictive control (LEMPC) design that can economically operate nonlinear descriptor systems while satisfying input and states constraints. To guarantee closed-loop stability and recursive feasibility of the proposed control design, a Lyapunov-based control law is introduced to characterize the stability region at which the LEMPC can maximize the process economics. Finally, a chemical batch process modeled by nonlinear differential algebraic equations (DAEs) is utilized to demonstrate the applicability of the proposed framework.

Fast and virtually exact quantum gate generation in U(n) via iterative Lyapunov methods

ABSTRACT A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let be the trajectory that is obtained in step , with . In step , is right-translated in order to displace to . The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating , and so on. A proof of the exponential convergence of to zero is provided for large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time . It presents excellent results for . The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.

Position Tracking Control of Tailsitter VTOL UAV With Bounded Thrust-Vectoring Propulsion System

This paper presents an adaptive position tracking controller for a special kind of tailsitter vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). The distinctive part of this VTOL aircraft is its thrust-vectoring propulsion system, which allows the aircraft to achieve long-range flight at high-efficiency. However, there are two main problems encountered in the control design for this aircraft in vertical mode. One problem is that the mass and inertia tensor are unknown parameters because of the fuel consumption. The other problem is that the thrust vector is bounded both in amplitude and orientation, leading to the boundedness of control input. In this paper, we use a complementary filter to estimate the mass and an adaptive law to compensate the error caused by inertia uncertainty. A Lyapunov-based control law with nested saturation function is proposed to solve the problem of input boundedness. Finally, the global asymptotic stability is proven for the simplified model and the effectiveness is illustrated through simulation.

Formation Control and Trajectory Tracking of Nonholonomic Mobile Robots

In this brief, we design Lyapunov-based control laws to achieve two multi-objective tasks for a network of open-loop unstable, nonholonomic mobile inverted pendulum (MIP) robots, using a connected undirected graph for inter-agent communication. Using the first protocol, translationally invariant formations are achieved along with the synchronization of attitudes and heading velocities to desired values. Using the second protocol, the robots move into a formation and asymptotically track a trajectory. The control laws are based on the kinematic model of the mobile robot, and control torques for the MIPs are extracted using a two-loop control architecture. Both the protocols guarantee boundedness of the linear heading velocity, which is necessary for the stability of the two-loop control architecture. The proposed control laws are experimentally validated on indigenously built MIP robots.

Robust Nonlinear Regulation of Limit Cycle Oscillations in UAVs Using Synthetic Jet Actuators

In this paper, a synthetic jet actuators (SJA)-based nonlinear robust controller is developed, which is capable of completely suppressing limit cycle oscillations (LCO) in UAV systems with parametric uncertainty in the SJA dynamics and unmodeled external disturbances. Specifically, the control law compensates for uncertainty in an input gain matrix, which results from the unknown airflow dynamics generated by the SJA. Challenges in the control design include compensation for input-multiplicative parametric uncertainty in the actuator dynamic model. The result was achieved via innovative algebraic manipulation in the error system development, along with a Lyapunov-based robust control law. A rigorous Lyapunov-based stability analysis is utilized to prove asymptotic LCO suppression, considering a detailed dynamic model of the pitching and plunging dynamics. Numerical simulation results are provided to demonstrate the robustness and practical performance of the proposed control law.

Robust Attitude Control with Improved Transient Performance

This paper aims to present a robust attitude control strategy with guaranteed transient performance. Firstly, a Lyapunov-based control law is designed to achieve high-performance attitude control in the absence of disturbance and parameter variation. The proposed control law uses small feedback gains to suppress the control torque at large attitude error, and increases those gains with the convergence of attitude error to accelerate the system response. The overshooting phenomenon is also avoided by imposing a restriction on the parameter selection. Then, the integral sliding mode control technique is employed to improve the robustness, where the Lyapunov-based control law is used as the equivalent control part. Theoretical analysis and simulation results verify the effectiveness of the proposed strategy.

A Lyapunov-based Adaptive Control Law for an Electromagnetic Actuator

Electromagnetic devices in industrial systems are utilised for problems of positioning and tracking control. A direct adaptive non-linear control framework for a non-linear electromagnetic actuator is presented in this paper. Starting from a Lyapunov approach a continuous control law is calculated for the actuator. After that, an approximated sampled final control law is proposed. The resulting control strategy is an adaptive one in which the main parameter depends on the variable inductance of the actuator itself. This aspect seems to be important to prevent saturation during the control phase.

Preparation of entanglement states in a two-spin system by Lyapunov-based method

A stable Lyapunov-based control law is proposed for entanglement states preparation in this paper. The mathematical model of a two-spin system is constructed in interaction picture. An average value of observable operator is selected as the Lyapunov function. This paper designs the observable operator to make sure that the target state is stable in Lyapunov sense. Then the authors analyze the convergence of the control system, and demonstrate that the Bell states are asymptotically stable under the control law designed. Compared with other methods, the control method proposed in this paper has the advantages of easy design and convergence to the target states. Numerical simulation experiments for the preparation of Bell states are implemented.

Unmanned Aerial Vehicle Coordination on Closed Convex Paths in Wind

I N THIS Note we consider the problem of motion coordination of an aerial robotic network in the presence of steady, uniform winds. We present application examples of multivehicle coordination and path planning methods. The control laws we employ are extensions of previous research results on particle coordination and minimum-time path planning in steady, uniform wind. In prior work most relevant to this Note, Lyapunov-based control laws are provided to drive a collective of vehicles to a symmetric pattern along a circular orbit [1], to coordinated patterns on convex curves [2], and to coordinated patterns in the presence of winds [3]. The development of minimum-time path planning algorithms is also relevant [4,5]. The vehicle coordination framework described here may serve as an intermediate layer between low-level vehicle control and highlevel mission planning. We assume that the control signal is the turn rate-of-change, which, in the case of a fixed-wing aircraft, may be controlled by regulating the bank angle.We also assume that the lowlevel controller (the autopilot) can execute the desired turn-rate commands. Similar vehicle models have been frequently used to design kinematic control laws to track targets with aerial vehicles [6– 9]. Coordinated steering laws were presented in [10], where it was shown that collective motion along parallel lines or around the same circle are the only relative equilibria if the particle steering laws depend only on relative positions and headings. Motivated by the need for coordination on curves of arbitrary shapes, Lyapunov-based control laws were presented in [11], in which relative arc lengths between particles were used for coordination, as opposed to relative phases. Leader–follower approaches have also been studied (see, for example, [12,13] and the references therein). The contributions of this Note are twofold: First, we extend an existing motion-coordination framework to the case in which the vehicles travel around strictly convex loops in the presence of a steady, uniform flowfield. Second, we present an approximation method that generates strictly convex curves from convex paths that may contain straight segments. The latter method is important to allow coordination on time-optimal paths, which frequently contain straight segments. This Note also includes simulation results focusing on two application examples: control-volume sampling and perimeter defense. In Sec. II we describe the motion-coordination algorithms. In Sec. III we present the curve-approximation method that enables coordinated control on approximately time-optimal paths. In Secs. IV and V we describe the application examples. Section VI summarizes this Note.

A time differences of arrival‐based homing strategy for autonomous underwater vehicles

AbstractA new sensor‐based homing integrated guidance and control law is presented to drive an underactuated autonomous underwater vehicle (AUV) toward a fixed target, in 3‐D, using the information provided by an ultra‐short baseline (USBL) positioning system. The guidance and control law is first derived at a kinematic level, expressed on the space of the time differences of arrival (TDOAs), as directly measured by the USBL sensor, and assuming the plane wave approximation. Afterwards, the control law is extended for the dynamics of an underactuated AUV resorting to backstepping techniques. The proposed Lyapunov‐based control law yields almost global asymptotic stability (AGAS) in the absence of external disturbances and is further extended, keeping the same properties, to the case where known ocean currents affect the motion of the vehicle. Simulations are presented and discussed that illustrate the performance and behavior of the overall closed‐loop system in the presence of realistic sensor measurements and actuator saturation. Copyright © 2009 John Wiley & Sons, Ltd.

A Sensor-Based Controller for Homing of Underactuated AUVs

A new sensor-based homing integrated guidance and control law is presented to drive an underactuated autonomous underwater vehicle (AUV) toward a fixed target, in three dimensions, using the information provided by an ultrashort baseline (USBL) positioning system. The guidance and control law is first derived using quaternions to express the vehicle's attitude kinematics, which are directly obtained from the time differences of arrival (TDOA) measured by the USBL sensor. The dynamics are then included resorting to backstepping techniques. The proposed Lyapunov-based control law yields global asymptotic stability in the absence of external disturbances and is further extended, keeping the same properties, to the case where constant known ocean currents affect the dynamics of the vehicle. Finally, a globally exponentially stable nonlinear TDOA and range-based observer is introduced to estimate the ocean current and uniform asymptotic stability is obtained for the overall closed-loop system. Simulations are presented illustrating the performance of the proposed solutions.

Control and Monitoring of a High Recovery Reverse Osmosis Desalination Process

Model-based control and monitoring such as feed-forward/feedback control, fault detection and isolation (FDI), and fault-tolerant control (FTC) techniques that utilize Lyapunov-based control laws are implemented on a high recovery reverse osmosis desalination plant model. A detailed mathematical model of a high recovery reverse osmosis plant is developed. This model incorporates the large spatial variations of concentration and flow rate that occur in membrane units during high recovery operation. Bounded nonlinear feedback and feed-forward controllers are developed and applied to this system. The application of these controllers with FDI and FTC is demonstrated in the context of a high recovery reverse osmosis process simulation. The first set of simulations demonstrates the ability to compensate for the effects of large time-varying disturbances in the feed concentration on specific process outputs with and without feed-forward control. The second set of simulations demonstrates the ability of FDI and F...

Direct Lyapunov-based control law design for spacecraft attitude maneuvers

Abstract A direct Lyapunov–based control law is presented to perform on–orbit stability for spacecraft attitude maneuvers. Spacecraft attitude kinematic equations are coupled, nonlinear, multi–input multi–output (MIMO), which baffles controller design. Orbit angular rates are taken into account in kinematic equations and influence of gravity gradient moments and disturbance moments on the spacecraft attitude in dynamic equations is considered to approach the practical environment, which enhance the problem complexity to some extent. Based on attitude tracking errors and angular rates' a Lyapunov function is constructed' through which the stabilizing feedback control law is deduced via Lie derivation of the Lyapunov function. The proposed method can deal with the case that the spacecraft is subjected to mass property variations or centroidal inertia matrix variations due to fuel assumption or flexibility, and disturbance moments, which shows the proposed controller is robust for spacecraft attitude maneuve...

Satellite Orbit Transfer and Formation Reconfiguration via an Attitude Control Analogy

A control law applicable to orbit transfer as well as formation reconfiguration has been developed via a Lyapunov function similar to that used for spacecraft attitude control. It uses feedback of kinematic and kinetic error states between the current and desired orbital frames, which are parameterized by Euler parameters, orbital radius and its time derivative, and orbital angular velocities. The efficacy of the control law is demonstrated with examples, including orbit transfers and formation reconfiguration, both with low-eccentricity and high-eccentricity reference orbits, in the presence of J2 perturbations. Further simplifications in the control law are introduced and it is shown that stability is not affected. A methodology is proposed to reduce control requirement by appropriate gain selection and control initiation at predefined locations. The equivalent impulse requirements obtained for the examples compare very favorably with analytically and numerically optimized figures presented in other published works. HE problem of orbital transfers of satellites by the use of con- tinuous, low-thrust control has been studied in great detail. Re- cent advances in the field of electric propulsion have made low-thrust propulsion an operational reality. Formation flying of spacecraft is also a relatively new area of research wherein control of relative mo- tion is a key element. Control of relative motion requires controlling each satellite's orbit precisely, with high regard for intersatellite dis- tances, collision hazards, and other operational constraints. The primary purpose of this paper is to develop a Lyapunov-based control law that has a large range of applicability, free from the influ- ence of singularities that arise from the more common descriptions of a satellite's orbit, such as orbital elements. Previous studies in the literature have addressed the problem of continuous-thrust orbit transfer using Lyapunov function-based feedback controllers. Ilgen 1 uses orbital element feedback for orbital transfers, using the clas- sical as well as equinoctial elements. Ilgen's quadratic Lyapunov function is based on only five of the orbital elements. Gurfil 2 uses classical orbital element feedback for orbital transfer and explores nonlinear controllability issues of the problem, and is able to prove by using Gauss variational equations that a spacecraft's orbital el- ements can be controlled by continuous control, except when the desired orbit is parabolic. Chang et al. 3 approach the problem of orbital transfer by feeding back the angular momentum and eccen- tricity vectors. References 1-3 provide excellent bases for the de- velopment of Lyapunov-function-based controllers for formation reconfiguration using orbital elements. Schaub and Alfriend 4 use the local Cartesian coordinates of the Chief/leader satellite and the differential orbital elements between the Deputy/follower satellite and the Chief as feedback. Schaub et al. 5 present two control laws, one in terms of mean elements, and another in terms of inertial co- ordinates, for formation control, with the added simplification of using mean elements in the presence of J2 perturbations arising due

Robust stabilization of nonlinear systems with parametric uncertainty

Presents a result on the robust stabilization of a class of nonlinear systems exhibiting parametric uncertainty. The authors consider feedback linearizable nonlinear systems with a vector of unknown constant parameters perturbed about a known value. A Taylor series of the system about the nominal parameter vector coupled with a feedback linearizing control law yields a linear system plus nonlinear perturbations. Via a structure matching condition, a Lyapunov-based control law is shown to exponentially stabilize the full system. The novelty of the result is that the linearizing coordinates are completely known since they are defined about the nominal parameter vector, and fewer restrictions are imposed on the nonlinear perturbations than elsewhere in the literature. >