Linear Quadratic Regulator Method Research Articles

Autonomous vehicle tracking control for a curved trajectory

Recently, research about trajectory tracking of autonomous vehicles has significantly contributed to the development of autonomous vehicle technology, particularly with novel control methods. However, tracking a curved trajectory is still a challenge for autonomous vehicles. This research proposes a state feedback linearization with observer feedback to overcome some difficulties arising from such a path. This approach suits a complex nonlinear system such as an autonomous vehicle. This method also has been compared with the linear-quadratic regulator (LQR) method. So, the goal of this research is to improve the control system performance of autonomous vehicles that are stable enough to navigate a curved path. Moreover, the study shows that the developed control law can track the curved path and solve existing problems. However, improvements are still necessary for the vehicle's performance and robustness.

A deep learning optimized LQR method for enhanced formation control with embedded systems

To achieve higher accuracy throughout the formation control processes and enhance precision in dynamic environments, particularly for the formation control of follower vehicles with embedded systems, this paper proposes a method and framework for vehicle formation control. An Ackermann-model based Linear Quadratic Regulator (LQR) controller is developed for lateral distance control and a Proportional-Derivative (PD) controller for longitudinal distance control. To enhance the efficacy of the LQR controller, the Deep Deterministic Policy Gradient Derivative (DDPG) method is introduced into the control system. The DDPG networks are trained in a simulation environment and can subsequently predict LQR parameters in real-time experiments. The practical application of the method is showcased, and the concluding remarks emphasize the potential and superior performance of our proposed formation control approach by experimental comparison with other controllers. This method can be implemented in small vehicles that possess limited computational resources and is also suitable for scenarios requiring dynamic motion control with higher tracking accuracy and stability.

Indirect Measurement of Variables in a Heterogeneous Reaction for Biodiesel Production.

This research focuses on the development of a state observer for performing indirect measurements of the main variables involved in the soybean oil transesterification reaction with a guishe biochar-based heterogeneous catalyst; the studied reaction takes place in a batch reactor. The mathematical model required for the observer design includes the triglycerides' conversion rate, and the reaction temperature. Since these variables are represented by nonlinear differential equations, the model is linearized around an operation point; after that, the pole placement and linear quadratic regulator (LQR) methods are considered for calculating the observer gain vector L(x). Then, the estimation of the conversion rate and the reaction temperature provided by the observer are used to indirectly measure other variables such as esters, alcohol, and byproducts. The observer performance is evaluated with three error indexes considering initial condition variations up to 30%. With both methods, a fast convergence (less than 3 h in the worst case) of the observer is remarked.

Double-loop LQR depth tracking control of underactuated AUV: Methodology and comparative experiments

In this paper, a double-loop depth tracking control scheme based on linear quadratic regulator (LQR) is proposed for underactuated autonomous underwater vehicles (AUV). The motivation is to address the problem that the classic single-loop LQR depth controller which is easy to use in practice yet cannot effectively overcome the underactuated AUV features and complex disturbances. First, a double-loop depth control framework is proposed which divides complex disturbances into outer-loop kinematic disturbances and inner-loop dynamic disturbances. Second, the adaptive line-of-sight (ALOS) guidance law is designed in the outer loop to account for the underactuated characteristics. The depth tracking error is converted to the desired pitch and the kinematic disturbance caused by the unknown angle of attack is compensated. Furthermore, based on the linear model in the inner-loop, a concise LQR method is designed to track the desired pitch. Meanwhile, an extended state observer (ESO) is introduced to estimate dynamic disturbances such as model nonlinearity, uncertainty and external environmental disturbances. Finally, the proposed scheme is validated by tank test and compared with the single-loop LQR method. The experiment results show that the proposed double-loop depth tracking method can effectively overcome complex disturbances and the depth tracking accuracy is increased by 83%.

Online Adaptive Dynamic Programming for Optimal Self-Learning Control of VTOL Aircraft Systems With Disturbances

In this paper, a novel online adaptive dynamic programming control algorithm is developed to solve optimal control problems for Vertical Take Off and Landing (VTOL) aircraft systems. Considering the input disturbances of the VTOL systems and adding perturbation interference term to the value function, a robust policy iteration-based ADP method will be introduced to obtain the optimal controller with an approximate optimal control approach. The convergence property is developed to ensure the value function converges to a finite neighborhood of the optimal one. The uniform ultimate boundedness (UUB) of the iterative errors will be strictly proved by Lyapunov stability theory. Finally, we will give mathematical simulation results with the developed method. Compared with sliding mode control (SMC) and linear quadratic regulator (LQR) methods, the simulation results illustrate better overall performance of the novel method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —As an effective tool to solve the optimal control problem of complex nonlinear systems, adaptive dynamic programming (ADP) can effectively overcome the computational problems caused by “curse of dimensionality”. VTOL aircraft system is a typical strongly coupled, underactuate, non-minimum phase system, and the sudden random disturbances will make the control task more challenging. Aiming at the particularity of VTOL system, a new online policy iteration-based ADP algorithm is developed to obtain optimal control with an approximate optimal control strategy in this paper. Decoupling and disturbances can be effectively avoided. convergence will be analyzed to make sure the value function converge to a finite neighborhood of the optimal value function. The UUB property of the system will be proved by the Lyapunov approach.

Research on wheel-legged robot based on LQR and ADRC

The traditional two-wheeled self-balancing robot can travel quickly in a flat road environment, and it is easy to destabilize and capsize when passing through a bumpy road. To improve the passing ability of a two-wheeled robot, a new wheel-legged two-wheeled robot is developed. A seven-link leg structure is proposed through the comprehensive design of mechanism configuration, which decouples the balanced motion and leg motion of the robot. Based on the Euler–Lagrange method, the dynamic model of the system is obtained by applying the nonholonomic dynamic Routh equation in the generalized coordinate system. The robot’s state space is divided according to the robot’s height, and the Riccati equation is solved in real-time by the linear quadratic regulator (LQR) method to complete the balance and motion control of the robot. The robot leg motion control is achieved based on the active disturbance rejection control (ADRC) way. A robot simulation model is built on Recurdyn to verify the algorithm’s feasibility, and then an experimental prototype is built to demonstrate the algorithm’s effectiveness. The experimental results show that the control method based on LQR and ADRC can make the robot pass through the bumpy road.

Generating Symmetric Periodic Orbits and Their Optimal Maintenance in a Binary Asteroid System

In this paper, generating symmetric periodic orbits around a binary asteroid and optimal maintenance of spacecraft on these periodic orbits are investigated. Due to the irregular shape of asteroids, numerical methods are needed to generate periodic orbits. In this regard, the 32 families of symmetric periodic orbits were calculated by grid search and shooting methods utilizing the restricted three-body problem as a basic model. The irregularity of the shape of asteroids and solar radiation pressure are considered perturbations. These perturbations should be compensated by a thruster with a continuous thrust through the continuous control strategy, i.e., the linear quadratic regulator (LQR) method. The effect of different matrix coefficients in the performance of the LQR method is examined. The results show that the LQR method could appropriate control spacecraft on periodic orbits with specular shapes. Also, the practical feasibility of a continuous thrust control strategy is discussed.

Damping control of polodes, inertia and natural frequencies: Theory and application to automotive suspensions

This paper shows how tunable dampers can help control the instant centre of rotation of a 2D rigid body and its polode in planar motion, which in turn implies that the inertia tensor can also be controlled. For mechanisms equipped with some elasticity the results show that damping can also control their natural frequencies. The foundation of a general theory to control the polode is presented, exploring the chance of an optimal control formulation of the problem via a variational control principle, approached by the LQR (Linear Quadratic Regulator) method, after a suitable linearization. Application to automotive suspension linkages is presented that demonstrates the control of the instant roll centre and axis and consequently its instant roll vibration frequency to optimize the response, when excited by lateral inertia forces.

Research on the High-Speed Collision Avoidance Method of Distributed Drive Electric Vehicles

High-speed autonomous emergency collision avoidance for intelligent vehicles is a challenging task. Many existing studies have not considered the effect of the road adhesion coefficient in the vehicle high-speed collision avoidance planning layer. Additionally, in lateral control, the effect of strong coupling of vehicle motion on the system model is not considered. This effect increases the uncertainty in the collision avoidance process and correlates with driving safety. In this regard, this paper conducted relevant studies. First, a distributed drive electric vehicle (DDEV) with controlled longitudinal speed was selected as the research platform and derived the transformed square-root cubature Kalman filter (T-SCKF) algorithm with cubature point sampling rule transformation to estimate the road adhesion coefficient. Second, the road adhesion coefficient information was used to calibrate the following vehicle safety distance model and the lane change collision avoidance lateral acceleration information was used to determine the vehicle autonomous emergency collision avoidance trajectory and to optimize the planning decision of the vehicle collision avoidance process. Last, the DDEV was laterally controlled to complete the tracking of the high-speed emergency collision avoidance trajectory by the adaptive model predictive control (Adaptive MPC) method with real-time updating of the system model. The relevant validation shows that the T-SCKF algorithm proposed in this paper can effectively estimate the road adhesion coefficient. The tracking control of the high-speed collision avoidance trajectory by the adaptive MPC method has better tracking accuracy and stability than traditional MPC and linear quadratic regulator (LQR) methods.

Optimal LQR Controller Methods for Double Inverted Pendulum System on a Cart

Most of the systems in our lives are inherently nonlinear and unstable. In control problems in the field of engineering, the aim is to define the control laws that maximize the operating efficiency of these systems under diverse security coefficients, and constraints and minimize error rates. This study aimed to model and optimally control a Double-Inverted Pendulum System on a Cart (DIPSC). A DIPSC was modeled using the Lagrange-Euler method, and classical and optimal Linear Quadratic Regulator (LQR) control methods were designed for the control of the system. The purpose of the designed controllers is to keep the arms of the double inverted pendulum on the moving cart vertically in balance and to bring the cart to the determined balance position. The critically important Q and R parameters of the LQR control technique that is one of the optimal control techniques were obtained using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Grey Wolf Optimization (GWO) algorithms. The DIPSC system was checked using classical LQR and optimal LQR methods. All obtained results are given graphically. The proposed methods are presented and analyzed in tabular form using Settling time and Mean-Square-Error (MSE) performance criteria.

Path-following control method for articulated vehicles based on dual heuristic programming

A path-following control method based on dual heuristic programming (DHP) is proposed to address the problem whereby unmanned articulated vehicles need to set different controller parameters to track complex routes and cannot adapt to complex routes. First, the path-following control system structure is designed based on the articulated vehicle experimental platform, and the error model is derived based on the kinematic model of the articulated vehicle. Second, the payoff function is designed considering the error and stability indices, and the actor and critic of the path-following control method based on the DHP algorithm are approximated using a multilayer feedforward neural network. Finally, the path-following quality of the method is verified using simulations and real vehicle tests and compared with the conventional pure tracking and linear-quadratic regulator methods. The results show that the DHP-based method is able to follow complex reference paths and obtain better control results than the traditional methods without iterative tuning of the controller parameters, which improves the adaptiveness of the articulated vehicle path-following control method to complex environments.

Toward Design and Implementation of Self-Balancing Robot Using Deep Learning

In the Internet of Things (IoT) era, an immense amount of sensing devices are obtained and produce various sensory data over time for a wide range of disciplines and applications. These devices will result in significant, fast, and real-time data streams based on the utilization characteristics. Utilizing analytics over such data streams to identify new information, model future insights, and make control decisions is a necessary process that makes IoT a worthy paradigm for businesses and a quality-of-life improving technology. This paper presents a study of digital agriculture and its significance in terms of the application of an IoT-based device — a two-wheeled self-balancing robot — followed by a thorough procedural explanation of the development of the device, which begins with the mathematical modeling of the system through the Euler–Lagrange method to obtain the equations of motion for the same and linearize the equation to define the control method to be used to balance the robot structure, all based on the concept of the inverted pendulum. Then paper discusses the suitable and the most efficient control method, which is the linear quadratic regulator (LQR), for these robots. Then deep learning-based LQR (DL-LQR) method is implemented in the robots performing the algorithm to balance it successfully.

Uncertain optimal attitude control for space power satellite based on interval Riccati equation with non-probabilistic time-dependent reliability

Considering the multi-source uncertainties and disturbances in the attitude dynamics of space power satellite (SPS), an interval dynamic model of SPS is proposed based on the non-probabilistic theory, which is controlled by a novel uncertainty-based linear–quadratic regulator (LQR) method with time-dependent reliability. Based on the interval analysis method, an interval time-varying attitude dynamic model of SPS for sun-pointing is established considering the uncertain inertia and initial states as unknown-but-bounded parameters. Feedforward and feedback controllers are designed to offset the disturbance and control attitude, respectively. An interval Riccati equation-based optimal control method is proposed to develop a novel uncertainty-based LQR method. Non-probabilistic time-dependent reliability is constituted based on the interval process and first passage theory to assess attitude states. An uncertain multi-objective optimization of attitude control with reliability constraints is proposed to realize the uncertain LQR method. Verified by a numerical example of SPS, the proposed method takes only 0.6‰∼0.8‰ time consumption of 10,000 Monte Carlo simulations while maintaining a certain accuracy.

Decentralized State Estimation-Based Optimal Integral Model Predictive Control of Voltage and Frequency in the Distribution System Microgrids

The power distribution system is undergoing a shift from centralized to decentralized operation due to the large-scale integration of distributed energy resources (DER). To increase the scalability and reduce the dependency on communication networks, there is a strong need for a decentralized control framework that can maintain the nominal voltage and frequency in a microgrid. In this article, a novel decentralized state estimation-based optimal voltage and frequency restoration strategy is proposed. The framework computes the optimal integral model predicted control reference signal to the primary controller, that is unaffected by the local measurement noise, hence, ensuring stable power-sharing among the DERs. The proposed framework is built on a first-order DER model and a local/global measurements-based local state estimation technique, facilitating deployability to the grid edge devices. The capabilities of the proposed framework are tested and validated on a real-time simulator with a full system dynamic model considering an islanded section of the IEEE 123 node distribution network for the various grid events. It is observed that the framework offers significant transient performance improvement in comparison to the Linear Quadratic Regulator method, with around 70% faster voltage restoration when the communication latency is considered.

Design of LQR-based FLC for the Optimal Regenerative Braking Controlling of Solar PV-based Electric Vehicle System

The article presents an analysis of the optimal control of speed and regenerative braking in an electric vehicle (EV). The presence of the continuous generating power is one of the major challenges for battery charging of electric vehicle. In order to meet the demand of continuous power, electric vehicles are driven by solar PV arrays. Initially, maximum power extraction (MPPT) is assessed from a solar photovoltaic (PV) array using the P&O method. The outputs of the solar PV array are integrated with electric vehicles. Such integration creates the problem of maintaining the power quality. The existing methods also address the issue of optimal speed and torque control with high transients and more settling time. In this article, an electric vehicle is modeled first and then operated in regenerative braking mode. To address the shortcomings of existing methods, the linear quadratic regulator (LQR) method is used; then, fuzzy logic controller (FLC)-based LQR is used for optimal speed and torque control during electric vehicle regenerative braking. In terms of reduced settling time and improved peak overshoot, optimal speed and regenerative torque of electric vehicles are better realized with LQR-based FLCs than with LQR controllers. In addition to this, stopping speed characteristics with various electrical parameters (voltage, current, and flux) have been analyzed with different methods in which LQR-based FLC shows its dominance to attain zero speed in regenerative mode with more sharp characteristics.

Experimental studies of linear quadratic regulator (LQR) cost matrices weighting to control an accurate take-off position of bicopter unmanned aerial vehicles (UAVs)

Controller design for airplane flight control is challenged to achieve an optimum result, particularly for safety purposes. The experiment evaluated the linear quadratic regulator (LQR) method to research the optimal gain of proportional-integral-derivative (PID) to hover accurately the bicopter model by minimizing error. The 3 degree of freedom (DOF) helicopter facility is a suitable bicopter experimental simulator to test its complex multiple input multiple output (MIMO) flight control model to respond to the challenge of multipurpose drone control strategies. The art of LQR setting is how to search for appropriate cost matrices scaling to optimize results. This study aims to accurately optimize take-off position control of the bicopter model by investigating LQR cost matrices variation in actual experiments. From the experimental results of weighted matrix variation on the bicopter simulator, the proposed LQR method has been successfully applied to achieve asymptotic stability of roll angle, although it yielded a significant overshoot. Moreover, the overshoot errors had good linearity to weighting variation. Despite that, the implementation of cost matrices is limited in the real bicopter experiment, and there are appropriate values for achieving an optimal accuracy. Moreover, the unstable step response of the controlled angle occurred because of excessive weighting.

A semi-active control system in coupled buildings with base-isolation and magnetorheological dampers using an adaptive neuro-fuzzy inference system

Connecting two buildings has been proved as an effective method of structural control for alleviating seismic responses. Researchers have proposed that two adjacent buildings through supplemental energy dissipating devices to mitigate the buildings’ responses. Numerous researchers have proposed various methods: active, passive, and semi-active control strategies. In Japan, some applications of coupled buildings control have been successfully implemented by utilizing passive and active control technology. Magnetorheological (MR) dampers have been identified as semi-active devices that can be used to reduce the vibration of the seismic structures during various types of ground motions. They can offer the adaptability of active devices, stability, and reliability of passive devices. Nevertheless, one of the difficulties in application of the MR dampers is the development of the appropriate control algorithms. Accordingly, this study presents the implementation of the adaptive neuro-fuzzy inference system (ANFIS) controller for earthquake hazard mitigation under coupled buildings control system with base-isolated building connecting to the free wall by MR dampers. The ANFIS whose training data is based on the Linear Quadratic Regulator (LQR) method is conducted to modify the parameters of the fuzzy logic controller and optimize the fuzzy rules. The performance of MR dampers is evaluated under seismic response. It is compared under four methods, including passive-off, passive-on, and two semi-active control strategies: ANFIS and LQR. Besides, various types of feedback of the ANFIS operated as two-input single output feedback system are investigated to assess the performance of the developed control scheme for structural vibration control. The numerical simulation results show that the proposed semi-active control system consisting of coupled buildings system and MR dampers by utilizing ANFIS can be effective in mitigating seismic responses of structures.

Implementation and Stabilization of a Quadcopter Using Arduino and the Combination of LQR and SMC Methods

This paper presents a combination of Linear Quadratic Regulator (LQR) and Sliding Mode Control (SMC) methods to control a four-rotor unmanned aerial vehicle (UAV) that takes off and lands vertically (VTOL). Although controlling UAVs is difficult due to their highly nonlinear characteristics, the controller successfully controlled and stabilized the quadcopter in altitude and attitude by combining the advantages of the nonlinear and linear controllers. The Newton-Euler method is employed to build the dynamic model of the quadcopter, which is divided into two subsystems: the under-controlled subsystem and the fully actuated subsystem. The entire controller model was demonstrated in MATLAB/Simulink, and results demonstrating the controller's performance in various scenarios were obtained.

Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode

To resolve the problem of high-precision trajectory tracking control under interference conditions in a missile’s mid-guidance phase, according to the constructed nominal trajectory, an improved adaptive high-order sliding mode trajectory tracking controller (AHSTC) is proposed. In this method, the open-loop nominal trajectories are established according to the nonlinear programming and Gaussian pseudospectra method. A high-precision trajectory tracking controller is developed by designing a nonlinear sliding mode surface and an adaptive high-order sliding mode approaching law combined with the trajectory tracking nonlinear error model. To verify the effectiveness and superiority of the proposed method, analysis and simulation are carried out through the example of a missile mid-guidance phase tracking control. Compared to the linear quadratic regulator (LQR) and active disturbance rejection controller (ADRC) method, the simulation results show that the proposed AHSTC method shows faster convergence and improved tracking effect. Therefore, the proposed AHSTC method has a good results and engineering application value.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.