Optimal Linear Quadratic Regulator Research Articles

Improved Model Predictive Speed Control of a PMSM via Laguerre Functions

This paper proposes a model predictive speed control strategy for a surface-mounted permanent magnet synchronous motor by applying Laguerre functions. The model predictive controller (MPC) incorporates an integrator. A quadratic programming procedure is applied to solve the constrained optimization problem online. The paper also provides a solution for stability. The performance efficiency of the proposed scheme is validated by comparing the results with the performance of an optimal linear quadratic regulator, conventional state-space model predictive control, and a simple MPC algorithm with integral action. Extensive simulation results confirm the efficacy of the proposed scheme, showing that it achieves good steady-state performance while maintaining a fast dynamic response.

Observer-based dynamic event-triggered control for distributed parameter systems over mobile sensor-plus-actuator networks

Abstract The objective of this paper is to develop a policy of observer-based dynamic event-triggered state feedback control for distributed parameter systems over a mobile sensor-plus-actuator network. It is assumed that the mobile sensing devices that provide spatially averaged state measurements can be used to improve state estimation in the network. For the purpose of decreasing the update frequency of controller and unnecessary sampled data transmission, an efficient dynamic event-triggered control policy is constructed. In an event-triggered system, when an error signal exceeding a specified time-varying threshold it indicates the occurrence of a typical event. The global asymptotic stability of the event-triggered closed-loop system and the boundedness of the minimum inter-event time can be guaranteed. Based on the linear quadratic optimal regulator, the actuator selects the optimal displacement only when an event occurs. A simulation example is finally used to verify the effectiveness of such a control strategy can enhance the system performance.

Optimal Control of Hybrid Photovoltaic/Thermal Water System in Solar Panels Using the Linear Parameter Varying Approach

During photovoltaic (PV) conversion in solar panels, a part of the solar radiation is not converted to electricity by the cells, producing heat that could increase their temperature. This increase in temperature deteriorates the performance of the PV panel. In this paper, a hybrid PV/thermal (PV/T) water system is proposed to mitigate this problem. This system combines a PV panel and a thermal collector. In this paper, we focused on the modeling and control of this hybrid system in the linear parameter varying (LPV) framework. An optimal linear quadratic regulator (LQR) is proposed to control the PV cell temperature around an optimal value that maximises electricity generation. Since the system model is nonlinear, an optimal LQR gain-scheduling state-feedback control approach based on an LPV representation of the nonlinear model is designed using the Linear Matrix Inequality (LMI) method. The goal is to obtain the maximum electrical power for each solar panel. Since a reduced number of sensors is available, an LPV Kalman filter is also proposed to estimate the system states required by the state-feedback controller. The obtained results in a laboratory setup in simulation are used to assess the proposed approach, showing promise in terms of control performance of the PV/T system.

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.

Optimal analytical cascade control for FOPDT systems

This paper proposes a novel control strategy based on optimal Linear Quadratic Regulator (LQR) design applied to cascade systems with stable, integrative, and unstable First Order plus Dead-Time (FOPDT) outer loop models. The secondary (inner) and primary (outer) loop controller designs are performed using the Simplified Filtered Smith Predictor (SFSP). The optimal analytical design is derived from a novel strategy where the pole of the internal loop is placed so that the LQR criterion can be achieved with a simple tuning of the external loop. In contrast to related work, the secondary controller is designed using an SFSP control structure to nominally achieve optimal LQR performance and accelerated rejection of disturbances acting on the inner-loop process. To illustrate the usefulness of the proposed design, we evaluated the strategy in different scenarios, showing the optimality and performance of the analytical design.

AN Optimization control Techniques for Aircraft Yaw Control Lateral Dynamics

Presently, advanced control theory plays a major role in the aircraft system for the control of Yaw, Roll and Pitch angles, which was very much necessary for the stabilization of aircraft system. Following the works of the control system community for aircraft applications, this paper concentrates on the control of Yaw angle by designing various optimal Linear Quadratic Regulator (LQR) controllers using standard existing tools. The optimal controllers are designed for the aircraft, tested and validated using MATLAB/SIMULINK environment, and the results are compared with different system conditions to select the best optimal LQR feedback controllers in future aircraft control systems.

Stability Analysis of Lane-Keeping Assistance System for Trucks under Crosswind Conditions

To enhance the control accuracy of lane-keeping assistance systems for trucks encountering crosswind-induced lateral deviations to improve the lateral stability of the vehicle, this study proposes a control strategy based on a linear quadratic regulator (LQR) using a path-tracking preview model. First, the lateral deviation is calculated using the path-tracking preview model. Then, an observer for the vehicle’s sideslip angle is designed using a vehicle lateral tracking deviation model and a Kalman filter controller, and this is used to solve the deviation of the sideslip angle. Finally, a feedforward controller is designed based on the LQR controller and a linear two-degrees-of-freedom vehicle model to eliminate steady-state errors arising from LQR optimization, thereby obtaining the steering angle of the vehicle when subjected to crosswind conditions. Comparing the test results of the sideslip angle, yaw rate, and lateral acceleration demonstrates that this strategy effectively improves the control accuracy of lane-keeping under crosswind conditions. The proposed method is validated through hardware-in-the-loop experiments on a test bench, yielding results consistent with simulations.

SIMULATION OF OPTIMAL SPEED CONTROL FOR A DC MOTOR USING LINEAR QUADRATIC REGULATOR (LQR)

This paper describes DC motor speed control based on optimal Linear Quadratic Regulator (LQR) technique. Controller's objective is to maintain the speed of rotation of the motor shaft with a particular step response.The controller is modeled in MATLAB environment, the simulation results show that the proposed controller gives better performance and less settling time when compared with the traditional PID controller.

Maximum power tracking for wind energy conversion systems via a high-order optimal disturbance observer-based LQR without a wind speed sensor

A high-order optimal disturbance observer (HOODO) is introduced to precisely estimate the aerodynamic torque and the variable wind speed; and consequently calculate the optimal speed of the generator without measuring the wind speed. As the immeasurable wind speed is varying fast, the aerodynamic torque, which is considered as a disturbance, is also changed fast, then the conventional assumption that the disturbance is slowly-varying (i.e., its first-time derivative is zero), is not applicable. The proposed HOODO design considers the fast and stochastic characteristics of the wind speed by relaxing aforementioned conventional assumption. Moreover, via the linear optimal control theory, the parameters of the HOODO are tuned systematically by proper selection of the elements of the diagonal weighting matrices. In this study, a compromise between the observer’s convergence rate and the ability to Gaussian noise suppression is also considered. This helps to solve the difficulties of many existing observers regarding the gain selection algorithms. Also in the article, for the first time, a control scheme is considered in combination with the proposed HOODO and the LQ controller. To maximize a power captured from the wind, a linear-quadratic regulator (LQR) is utilized to maintain the angular speed of the generator at the optimal speed reference. A stability analysis of the designed control scheme is also discussed. Comparative results of the HOODO under different orders as well as other observers are given to prove the effectiveness of the introduced estimation method. The obtained simulation results reveal that the HOODO has a superior estimation of disturbance. All the simulations are carried out in the MATLAB/Simulink software.

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.

Linear quadratic optimal regulation for multiplicative noise systems with special terminal penalty

This paper focuses on the linear quadratic optimal control problem for stochastic systems with a special terminal penalty. The specialty lies in that the terminal penalty is a quadratic term over the conditional expectations of the terminal state. The essential difficulty lies in an additional measurability constraint on the input induced by the conditional expectation term, which is shared by the control problems for the stochastic system with delays, rational expectations problems, control problems involving conditional expectation terms, asymmetric information control and so on. A new idea, dynamic programming combined with orthogonal decomposition, is proposed to deal with this problem. The orthogonal decomposition eliminates the additional measurability constraint on the input. The solvability condition and the optimal controller are obtained by developing generalised Riccati equations. Numerical experiment is provided to support the achieved results.

Singular Optimal Linear Quadratic Regulator of Switched Systems

In this paper, we will investigate the singular linear quadratic regulator (LQR) problem of switched linear system in finite time horizon. The proposed method is transforming into a switched LQR problem by adopting linear transformation. Next, we adopt an embedding transformation method to convert the switched LQR problem to a traditional optimal control problem, so the bang-bang-type solution of the embedded optimal control problem in finite time is the optimal solution to the switched LQR problem. The switching sequence of modes and the switching instants can be calculated by solving a closed-form optimal switching condition. The optimal state feedback control law is determined simultaneously. Then, by solving a sequence of Riccati equation, we find some conditions that ensure switched LQR problem can be convert to the singular LQR problem. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.

Optimal Linear-Quadratic Regulator for a Stochastic System under Mutually Inverse Time Preferences in the Cost

We investigate a long-run behavior of a linear stochastic system. It is assumed that the quadratic cost includes a time-varying function and its multiplicative inverse. Such a specification reflects the fact that time preferences used by agents to assess different types of losses evolve in opposite directions. We consider the case when priority is set for the losses associated with state deviations. The optimal control law is derived with respect to extended long-run average cost criteria. We provide conditions for the existence of an alternative control strategy, which is also optimal and is based on a solution of an algebraic Riccati equation.

Adaptive cooperation of optimal linear quadratic regulator and lumped disturbance rejection estimator-based tracking control for robotic manipulators

Adaptive cooperation of optimal linear quadratic regulator and lumped disturbance rejection estimator-based tracking control for robotic manipulators

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.

Guaranteed Stability Margins for Decentralized Linear Quadratic Regulators

It is well-known that linear quadratic regulators (LQR) enjoy guaranteed stability margins, whereas linear quadratic Gaussian regulators (LQG) do not. In this letter, we consider systems and compensators defined over directed acyclic graphs. In particular, there are multiple decision-makers, each with access to a different part of the global state. In this setting, the optimal LQR compensator is dynamic, similar to classical LQG. We show that when sub-controller input costs are decoupled (but there is possible coupling between sub-controller state costs), the decentralized LQR compensator enjoys similar guaranteed stability margins to classical LQR. However, these guarantees disappear when cost coupling is introduced.

Revisiting LQR Control From the Perspective of Receding-Horizon Policy Gradient

We revisit in this paper the discrete-time linear quadratic regulator (LQR) problem from the perspective of receding-horizon policy gradient (RHPG), a newly developed model-free learning framework for control applications. We provide a fine-grained sample complexity analysis for RHPG to learn a control policy that is both stabilizing and -close to the optimal LQR solution, and our algorithm does not require knowing a stabilizing control policy for initialization. Combined with the recent application of RHPG in learning the Kalman filter, we demonstrate the general applicability of RHPG in linear control and estimation with streamlined analyses.

Optimal linear-quadratic regulator for a stochastic system under mutually inverse time preferences in the cost

Проводится анализ линейной стохастической системы управления в долгосрочном периоде. Предполагается, что квадратичный целевой функционал содержит взаимно обратные множители, зависящие от времени. Такая специфика служит отражением разнонаправленных временных предпочтений субъектов по оценке разных видов потерь. При этом рассматривается ситуация приоритетного учета издержек, относящихся к состоянию. Находится вид закона управления, оптимального по критериям обобщенных долговременных средних. Приводятся условия, при которых существует альтернативная оптимальная стратегия, определяемая на основе решения алгебраического уравнения Риккати.

A multi-step estimation approach for optimal control strategies of interconnected systems with weakly connected topology

This paper studies optimal linear quadratic regulation (LQR) problem of discrete-time interconnected systems (ISs) defined over a weakly connected graph. Subsystems in an IS share information based on the topology of the system. The main challenge of this work in comparison to the standard LQR problem, stems from that a subsystem may never acquire information from some other subsystems, due to the weakly connected topology. In this paper, a multiple-step estimation approach is proposed to analytically design the optimal controller for ISs with weakly connected topology. Also, the optimal value of the cost function is explicitly derived. Finally, the effectiveness of the proposed method is illustrated by simulations using a connected vehicle system.

Second-Order NonConvex Optimization for Constrained Fixed-Structure Static Output Feedback Controller Synthesis

For linear time-invariant systems, the design of an optimal controller is a commonly encountered problem in many applications. Among all the optimization approaches available, the linear quadratic regulator (LQR) methodology certainly garners much attention and interest. As is well known, standard numerical tools in linear algebra are readily available to determine the optimal static LQR feedback gain matrix when all the system state variables are measurable. However, in various certain scenarios where some of the system state variables are not measurable, and consequent prescribed structural constraints on the controller structure arise, the optimization problem can become intractable due to the nonconvexity characteristics that can then be present. In such cases, some first-order methods have been proposed to cater to these problems, but all of these methods, if at all successful, are limited to linear convergence. To speed up the convergence, a second-order approach in the matrix space is essential, with appropriate methodology to solve the linear equality constrained static output feedback (SOF) problem with a suitably defined linear quadratic cost function. Thus, in this article, an efficient method is proposed in the matrix space to calculate the Hessian matrix by solving several Lyapunov equations. Then, a new optimization technique is applied to deal with the indefiniteness of the Hessian matrix. Subsequently, through Newton’s method with linear equality constraints, a second-order optimization algorithm is developed to effectively solve the constrained SOF LQR problem. Finally, two numerical examples are described, which demonstrate the applicability and effectiveness of the proposed method.