Linear Quadratic Regulator Optimal Control Research Articles

Design of intelligent control for flexible linear double inverted pendulum based on particle swarm optimization algorithm

For the problem of control of flexible linear double inverted pendulum (FLDIP), considering that the method of obtaining LQR parameters through traditional manual trial-and-error is too cumbersome, and the basic particle swarm optimization algorithm has a slower convergence speed and is prone to a local optimal solution, in this paper, a control method is proposed that it can achieve autonomous optimization based on improved the particle swarm optimization (PSO) algorithm. Firstly, the mathematical model of the FLDIPs is given to analyse the stability of the system. Then, the PSO algorithm is improved from the perspective of dynamically adjusting inertia weights to optimize linear quadratic regulator (LQR) parameters. Finally, through simulation comparison with the LQR control effect obtained by the manual trial and error method, the feasibility of the improved PSO algorithm is verified. This method can not only improve the control performance of the system but also effectively overcome the problem of blindness in selecting Q for traditional LQR optimal control.

A unified study of the redundant control input problem with different conditions in the stabilizable case

This paper investigates the redundant control input problem in the sense of linear quadratic regulator optimal control. Increasing redundant control inputs leads to a decrease in the quadratic performance index. However, in some cases, this also increases the norm of the feedback matrix, resulting in actuator saturation. To solve the problem, a comparative analysis of the latest research results on the redundant control input problem under different conditions is first presented. Secondly, new tighter upper bounds of the positive (semi-)definite solution of the continuous algebraic Riccati equation (CARE) corresponding to the redundant control input problem are studied uniformly in the stabilizable case. Thirdly, based on them, a class of sufficient conditions to guarantee that the norm of the feedback matrix decreases is presented. Finally, numerical experiments demonstrate the effectiveness and superiority of our conclusions. Meanwhile, a simulation experiment demonstrates that systems with redundant control inputs can achieve good control performance.

Decoupled optimal control of 3D biped for human voluntary motion.

The sit-to-stand (STS) model from a biomechanical point of view is an enormously important subject, with motor controls simulating human intended behavior. Physiological motion-based biomechanical research is important for designing whole-body prosthetics and understanding physical disabilities. The control strategies for biomechanical models can effectively synergize with the central nervous system (CNS) to facilitate the desired movements of individuals with neurological disabilities. In this study, we present our novel 3D biped model by decoupling it into healthy and neurologically deficient joints. The developed 8-segment model (i.e., 2× feet, 2× shanks, 2× thighs, 1× pelvic, and 1× Head Arm Torso (HAT) segment) with 10 joints is decoupled into 6 healthy joints and 4 deficient joints. This decoupling mimics stroke patients or subjects with neuromuscular deficiency. This novel decoupling establishes through asymmetrical torques in frontal and sagittal plane joints on a bipedal design with one foot fixed and the other a sliding tilt joint. In this design, two decoupled controllers collaborate to stabilize the nonlinear model for biped STS transfer. Utilizing the xml files from SOLIDWORKS, the model is linearized in SIMSCAPE / SIMULINK. We further imply the Linear Quadratic Regulator (LQR) optimal controller design in MATLAB / SIMULINK for torques in both the sagittal and frontal planes, respectively, for six healthy and four deficient joints. We also comprehend the forward thrust velocity controls to pragmatically model the STS of stroke patients. This decoupling enhanced the overall stability of the system and simulated more relevant angular and velocity profiles for neurologically deficient substances.

Accurate solutions of interval linear quadratic regulator optimal control problems with fractional-order derivative

Accurate solutions of interval linear quadratic regulator optimal control problems with fractional-order derivative

Anti-deadzone adaptive fuzzy dynamic surface control for planar space robot with elastic base and flexible links

In order to combat the impact of the dead zone and reduce vibration of the space robot's elastic base and flexible links, the trajectory tracking and vibration suppression of a multi-flexible-link free-floating space robot system are addressed. First, the elastic connection between the base and the link is considered as a linear spring. Then the assumed mode approach is used to derive the dynamic model of the flexible system. Secondly, a slow subsystem characterizing the rigid motion and a fast subsystem relating to vibration of the elastic base and multiple flexible links are generated utilizing two-time scale hypotheses of singular perturbation. For the slow subsystem with a dead zone in joint input torque, a dynamic surface control method with adaptive fuzzy approximator is designed. Dynamic surface control scheme is adopted to avoid calculation expansion and to simplify calculation. The fuzzy logic function is applied to approximate uncertain terms of the dynamic equation including the dead zone errors. For the fast subsystem, an optimal linear quadratic regulator controller is used to suppress the vibration of the multiple flexible links and elastic base, ensuring the stability and tracking accuracy of the system. Lastly, the simulation results verify the effectiveness of the proposed control strategy.

Applications of the Linear Quadratic Regulator Optimal Control With Completely and Partially Observed Markovian Switching

Abstract We study optimal control problems where the dynamic system evolves according to a linear stochastic differential equation with (a) multiplicative and additive white noise, (b) a mixture of white noise, and (c) white noise and colored noise. The drift rate and the diffusion matrix of the linear dynamic system depend on a continuous-time Markov chain with finite state space that is (I) partially observed and (II) completely observed. Using the results of Wonham filter theory, we reduce the partially observed problems to one with complete observation. We solve the control optimal problems explicitly with the help of dynamic programing technique, and three applications are presented to illustrate our theoretical results.

Predictive control of seismic responses of a novel seismic isolation – non-seismic isolation composite structure system based on Levenberg-Marquardt algorithm

In order to reduce the seismic responses of adjacent structures, this paper proposes a novel seismic isolation – non-seismic isolation composite structure system, which is made up of a seismic isolation structure and a non-seismic isolation structure adjoined via controllers, and whose seismic responses are controlled with good effect by the principle of interacting control between the seismic isolation and non-seismic isolation structures. With a novel five-layer seismic isolation – non-seismic isolation composite structure system as an example, this paper builds a calculation model for the structure, derives the kinematic equations for the interacting control within this structural system, and adopts linear quadratic regulator (LQR) optimal control to analyze the control effect of this system and study the effect of time lag on the interacting control within this system. To address the issue of time lag in the interacting control between structures, an improved BP algorithm (Levenberg-Marquardt) is further adopted to set up a four-layer feedforward network to forecast the seismic responses of this novel composite structure system; the network structure is trained using the data acquired from LQR control under the action of the first 500 sampling points of the seismic motion taft, and the structure undergoes predictive control using the untrained seismic motion data of El Centro. The research results show that the interacting control of such a novel seismic isolation – non-seismic isolation composite structure system can achieve a good result, with the displacement response of the topmost layers of both structures decreasing by above 50 %; time lag may have an adverse effect on the control of the composite structure system, so that the structure’s seismic response increases by 20 % ∼ 40 %. The issue of time lag in interacting control can be addressed by adopting neutral network for predictive control, so that the structure’s seismic responses are 11 % ∼ 25 % lower than when it is not predicted. Levenberg-Marquardt algorithm demonstrates a strong learning ability and a curve approximation ability. The time history curves of the structure’s seismic responses under predictive control of this algorithm are highly overlapped with the time history curves under LQR optimal control with zero time lag.

Parametric Dynamic Distributed Containment Control of Continuous-Time Linear Multi-Agent Systems with Specified Convergence Speed

This paper focuses on the distributed containment control of continuous-time linear multi-agent systems (MASs) with multiple leaders over fixed topology. A parametric dynamic compensated distributed control protocol is proposed in which both the information from the observer in the virtual layer and actual adjacent agents are employed. The necessary and sufficient conditions of the distributed containment control are derived based on the standard linear quadratic regulator (LQR). On this basis, the dominant poles are configured by using the modified linear quadratic regulator (MLQR) optimal control and Geršgorin's circle criterion, hence the containment control with specified convergence speed of the MAS is achieved. Another main advantage of the proposed design is, in the case of virtual layer failure, by adjusting parameters the dynamic control protocol reduces to static, and the convergence speed can still be specified through the dominant pole assignment method combined with inverse optimal control. Finally, typical numerical examples are presented to demonstrate the effectiveness of theoretical results.

Linear Quadratic Gaussian Control for UAVs With Improved State Estimation Against Gyroscope and Accelerometer Biases

This paper proposes a linear quadratic Gaussian control for trajectory tracking of a quadrotor UAV. It involves implementing an optimal linear quadratic regulator control with integral action in an inner-outer loop control architecture. The full-state multi-rate extended Kalman filter generates the feedback required for the optimal control. Biases in gyroscope and accelerometer measurements are also incorporated into the Kalman filter to avoid degradation of the closed-loop response and to provide accurate state estimates. The proposed control architecture is tested on an experimental test bed consisting of a quadrotor UAV platform. The onboard inertial measurement unit, altimeter, and motion capture system provides the necessary measurements. The recorded results validate the performance of the proposed control scheme with improved state feedback generated through the extended Kalman filter.

Linear Quadratic Regulator Optimal Control of Two-Rotor Wind Turbine Mounted on Spar-Type Floating Platform

Abstract Interest is steadily growing for multi-rotor wind turbine concepts. This type of wind turbine offers a practical solution for scaling issues of large wind turbine components and for the reduction of costs associated with manufacturing, logistics, and maintenance. However, the literature lacks thorough knowledge of the dynamic performance of multi-rotor wind turbine concepts installed on floating platforms. Previous research studied the dynamic response of a two-rotor wind turbine concept mounted on a spar-type floating platform (2WT). Platform yaw motion is a significant dynamic factor directly caused by differential turbulence intensity experienced by the two hubs coupled with the distribution of thrust loads on the tower structure. Blade-pitch control analysis also showed how the 2WT yaw response is extremely sensitive to the control strategy employed. In this work, a linear quadratic regulator (LQR) is used to design an optimal controller for the 2WT prototype. Three LQR gain schedules corresponding to three operation regions are considered. An in-house tool for the dynamic analysis of multi-rotor floating wind turbines is used for linear state-space extraction and dynamic analysis. The control performance in different load conditions is assessed against the baseline OC3 proportional-integral (PI) control strategy and a PI-P control strategy in a previous article presented by the authors.

Modeling Dual-Drive Gantry Stages with Heavy-Load and Optimal Synchronous Controls with Force-Feed-Forward Decoupling

The application of precision dual-drive gantry stages in intelligent manufacturing is increasing. However, the loads of dual drive motors can be severely inconsistent due to the movement of heavy loads on the horizontal crossbeam, resulting in synchronization errors in the same direction movement of dual-drive motors. This phenomenon affects the machining accuracy of the gantry stage and is an critical problem that should be immediately solved. A novel optimal synchronization control algorithm based on model decoupling is proposed to solve the problem. First, an accurate physical model is established to obtain the essential characteristics of the heavy-load dual-drive gantry stage in which the rigid-flexible coupling dynamic is considered. It includes the crossbeam’s linear motion and rotational motion of the non-constant moment of inertia. The established model is verified by using the actual system. By defining the virtual centroid of the crossbeam, the cross-coupling force between dual-drive motors is quantified. Then, the virtual-centroid-based Gantry Synchronization Linear Quadratic Regulator (GSLQR) optimal control and force-Feed-Forward (FF) decoupling control algorithm is proposed. The result of the comparative experiment shows the effectiveness and superiority of the proposed algorithm.

Smart Active Vibration Control System of a Rotary Structure Using Piezoelectric Materials.

A smart active vibration control (AVC) system containing piezoelectric (PZT) actuators, jointly with a linear quadratic regulator (LQR) controller, is proposed in this article to control transverse deflections of a wind turbine (WT) blade. In order to apply controlling rules to the WT blade, a state-of-the-art semi-analytical solution is developed to obtain WT blade lateral displacement under external loadings. The proposed method maps the WT blade to a Euler–Bernoulli beam under the same conditions to find the blade’s vibration and dynamic responses by solving analytical vibration solutions of the Euler–Bernoulli beam. The governing equations of the beam with PZT patches are derived by integrating the PZT transducer vibration equations into the vibration equations of the Euler–Bernoulli beam structure. A finite element model of the WT blade with PZT patches is developed. Next, a unique transfer function matrix is derived by exciting the structures and achieving responses. The beam structure is projected to the blade using the transfer function matrix. The results obtained from the mapping method are compared with the counter of the blade’s finite element model. A satisfying agreement is observed between the results. The results showed that the method’s accuracy decreased as the sensors’ distance from the base of the wind turbine increased. In the designing process of the LQR controller, various weighting factors are used to tune control actions of the AVC system. LQR optimal control gain is obtained by using the state-feedback control law. The PZT actuators are located at the same distance from each other an this effort to prevent neutralizing their actuating effects. The LQR shows significant performance by diminishing the weights on the control input in the cost function. The obtained results indicate that the proposed smart control system efficiently suppresses the vibration peaks along the WT blade and the maximum flap-wise displacement belonging to the tip of the structure is successfully controlled.

Development and Evaluation of Control Architectures for a Mechanically Deployed Entry Vehicle

The NASA-funded Pterodactyl Project seeks to advance the current state of the art for atmospheric entry vehicles by developing novel guidance and control technologies for deployable entry vehicles (DEVs). This paper details the efforts of the Pterodactyl team to develop, implement, and evaluate different control hardware effectors on an asymmetric, mechanically deployed DEV. Multiple control effector designs are developed for a baseline vehicle including propulsive control systems (reaction control systems) and nonpropulsive control systems (aerodynamic control surfaces and internal moving masses). For each control system, state-feedback integral controllers based on linear quadratic regulator optimal control methods are designed to track the guidance commands of either the bank angle or the angle of attack and sideslip angle. Using a lunar return reference mission, a comparative analysis of the performance and maneuverability of the DEV using the different control system configurations is conducted. The aerodynamic control surface design is demonstrated to be the most suitable architecture, primarily due to its ability to achieve acceleration rates that supersede the other designs. Considering performance, maneuverability, and guidance tracking during entry, the aerodynamic control surface design is recommended for this DEV and chosen reference mission.

Missile Longitudinal Dynamics Control Design Using Pole Placement and LQR Methods

In high-maneuvering missile systems, with severe restrictions on actuator energy requirements, it is desirable to achieve the required performance with least actuation effort. Linear Quadratic Regulator (LQR) has been in literature for long and has proven it’s mettle as an optimal controller in many benign aerospace applications and industrial applications where the response times of the plant, in most cases, are seen to be greater than 10 seconds. It can be observed in the literature that LQR control methodology has not been explored enough in the tactical missile applications where requirement of very fast airframe response times are desired, typically of the order of milliseconds. In the present research, the applicability of LQR method for one such agile missile control has been critically explored. In the present research work, longitudinal dynamic model of an agile missile flying at high angle of attack regime has been established and an optimal LQR control solution has been proposed to bring out the required performance demanding least control actuator energy. A novel scheme has been presented to further optimise the control effort, which is essential in this class of missile systems with space and energy constraints, by iteratively computing optimal magnitude state weighing matrix Q and control cost matrix R. Pole placement design techniques, though extensively used in aerospace industry because of ease of implementation and proven results, do not address optimality of the system performance. Hence, a comparative study has been carried out to verify the results of LQR against pole placement technique based controller. The efficacy of LQR based controller over pole placement design techniques is successfully established with minimum control energy requirement in this paper. Futuristic high maneuvering, agile missile control design with severe space and energy constraints stand to benefit incorporating the controller design scheme proposed in this paper.

Summarisation, Simulation and Comparison of Nine Control Algorithms for an Active Control Mount with an Oscillating Coil Actuator

With the further development of the automotive industry, the traditional vibration isolation method is difficult to meet the requirements for wide frequency bands under multiple operating conditions, the active control mount (ACM) is gradually paid attentions, and the control algorithm plays a decisive role. In this paper, the ACM with oscillating coil actuator (OCA) is taken as the object, and the comparative study of the control algorithms is performed to select the optimal one for ACM. Through the modelling of ACM, the design of controller and the system simulations, the force transmission rate is used to compare the vibration isolation performance of the nine control algorithms, which are least mean square (LMS) adaptive feedforward control, recursive least square (RLS) adaptive feedforward control, filtered reference signal LMS (FxLMS) adaptive control, linear quadratic regulator (LQR) optimal control, H2 control, H∞ control, proportional integral derivative (PID) feedback control, fuzzy control and fuzzy PID control. In summary, the FxLMS adaptive control algorithm has the better performance and the advantage of easier hardware implementation, and it can apply in ACMs.

Investigation of the Structural Response of the MRE-Based MDOF Isolated Structure under Historic Near- and Far-Fault Earthquake Loadings

Fixed base structures subjected to earthquake forces are prone to various issues, such as the attraction of greater forces to structure, amplified accelerations to non-structural components, expensive design for better seismic performance, and so forth. Base isolation applied at the foundation of vulnerable structures is a radical bypass from the conventional approaches utilized by structural engineers. However, the practical implementation of passive base isolation is constrained by factors such as large displacements at isolation level, uplifting forces at isolators, and vulnerability to unpredictable and versatile earthquakes. This study is focused on the evaluation of the smart base isolation system under various harmonic and earthquake loadings. The proposed system employs a magnetorheological elastomer (MRE)—a class of smart materials, based on an adaptive isolation layer under the building structure for its vibration control. The building is idealized as a five-degree-of-freedom (DOF) structure with the mass lumped at each storey. The stiffness of the MRE isolation layer is adjusted using the linear quadratic regulator (LQR) optimal feedback control algorithm. A total of 18 simulations have been performed for the fixed base, passively isolated, and MRE-based isolated structures under a series of earthquake loadings of both a near-fault and far-fault nature for analyzing a total of 306 responses of the structures. The simulation results indicate that MRE-based isolation has significantly reduced all the responses compared to the passively isolated structure for both the near-fault and far-fault earthquake loadings. For harmonic loading, however, the passively isolated structure outperformed the MRE isolated structure in terms of storey drift and acceleration responses.

Dynamic Model and Intelligent Optimal Controller of Flexible Link Manipulator System with Payload Uncertainty

There is a high interest in research for using flexible link manipulators in industrial robots as flexible link manipulators are more advantageous than heavy and rigid link manipulators. However, flexible link manipulators still have a critical problem of less accuracy due to their tip vibration. Thus, this research contributes to this topic by obtaining a mathematical model and proposing an intelligent optimal controller for a single-flexible link manipulator with variable payload. The study developed the mathematical model of the single-flexible link manipulator using finite element method and Lagrange’s equation, the mathematical model of the flexible link manipulator has been validated with a SimMechamics model. The novel intelligent optimal controller is an integration of a fuzzy logic controller and an optimal linear quadratic regulator controller, the proposed intelligent optimal controller has the advantages of simplicity and effectiveness for position tracking and vibration suppression. The concept of integrating the fuzzy and linear quadratic regulator solves the problem of rules’ explosion of fuzzy control as only uses the minimum and active rules. The proposed controller has shown better position tracking performance and has demonstrated better effectiveness for vibration suppression of the flexible link manipulator than the linear quadratic regulator controller. Furthermore, the proposed controller is more robust than the linear quadratic regulator controller for dealing with uncertainties.

Robust Optimal Control for the Vehicle Suspension System With Uncertainties.

The suspension system is an important unit that affects vehicle performance, including handling stability and riding comfort. In order to improve vehicle performances, a novel robust optimal control strategy for the active suspension system in vehicles is proposed. To handle the parameter uncertainties and external random disturbances, the proposed optimal control strategy is designed by using the combination of the linear quadratic regulator (LQR) optimal control and sliding-mode control. First, by using the nominal model of the active suspension system without the uncertain parameters and the external random disturbances, an optimal control performance index is given, and the initial optimal controller is designed by using the LQR control strategy. Then, to handle the parameter uncertainties and the external disturbances, a robust optimal integral sliding-mode control strategy based on the initial optimal controller is designed, which not only can achieve the optimal control objective but also has good robustness to the uncertain parameters and external disturbances. By using the Lyapunov stability theory, stability analysis of the proposed robust optimal integral sliding-mode control strategy is performed. Finally, the collaborative simulation platform based on the Carsim and MATLAB/Simulink is developed. The simulation results illustrate the advantage of the proposed robust optimal control strategy for the suspension system.

Wind vibration control of stay cables using an evolutionary algorithm

In steel cable bridges, the use of magnetorheological (MR) dampers between butt cables is constantly increasing to dampen vibrations caused by rain and wind. The biggest problem in the actual applications of those devices is to launch a kind of appropriate algorithm that can effectively and efficiently suppress the perturbation of the tie through basic calculations and optimal solutions. This article discusses the optimal evolutionary design based on a linear and quadratic regulator (hereafter LQR) to lessen the perturbation of the bridges with cables. The control numerical algorithms are expected to effectively and efficiently decrease the possible risks of the structural response in amplification owing to the feedback force in the direction of the MR attenuator. In addition, these numerical algorithms approximate those optimal linear quadratic regulator control forces through the corresponding damping and stiffness, which significantly lessens the work of calculating the significant and optimal control forces. Therefore, it has been shown that it plays an important and significant role in the practical application design of semiactive MR control power systems. In the present proposed novel evolutionary parallel distributed compensator scheme, the vibrational control problem with a simulated demonstration is used to evaluate the numerical algorithmic performance and effectiveness. The results show that these semiactive MR control numerical algorithms which are present proposed in the present paper has better performance than the optimal and the passive control, which is almost reaching the levels of linear quadratic regulator controls with minimal feedback requirements.

Control system design for steering and depth subsystems of autonomous underwater vehicle

This work seeks to propose the development of a classical-optimized control design scheme for the steering and depth subsystems of autonomous underwater vehicle. Linearization methodological approach is carried out for acquiring the transfer functions to design the convenient P/PD/PID controllers where the gains are optimally adjusted utilizing a graphical tuning technique in LabVIEW. A low pass filter with additional alpha tuning parameter is combined with the derivative action to suppress the expected oscillations entirely. Furthermore, a comparative analysis study with linear quadratic regulator optimal control is established for testing and evaluating the suggested design approach. The effectiveness of the addressed controller strategy is appraised for compensating and rejection of several uncertainties such as fin saturation, angle limitation and sensor measurements noise as they are potentially affecting the system response. The validity of the linearized classical-optimized control design is substantiated with the nonlinear underwater vehicle model through a waypoint guidance scenario under LabVIEW simulation platform. The results revealed a significant improvement in transient and steady state responses and a remarkable enhanced stability for tracking the desired path to different designated waypoints.