Linear Quadratic Regulator Control Design Research Articles

LQR and Fuzzy-PID Control Design on Double Inverted Pendulum

Double inverted pendulum is a non-linear and unstable system. Double inverted pendulum can be stabilized in the upright position by providing control to the system. In this research we compare two types of controllers namely Linear Quadratic Regulator (LQR) and Fuzzy-PID. The objective is to determine the control strategy that provides better performance on the position of the cart and pendulum angle. We modelled the system which is then linearized and given control. From the simulation results, it is proven that LQR and Fuzzy-PID controllers have been successfully designed to stabilize the double inverted pendulum. However, when given a disturbance in the form of noise step, the LQR controller has not been able to achieve the desired reference for up to 20 seconds. In another hand, the Fuzzy-PID controller is able to achieve the desired reference after 8 seconds. Therefore, it can be concluded that the Fuzzy-PID controller when applied to the Double Inverted pendulum system has better performance than the LQR controller.

Towards High-Precision Quadrotor Trajectory Following Capabilities: Modelling, Parameter Estimation, and LQR Control

Abstract Quadrotor unmanned aerial vehicles (UAVs) are small, agile four-rotor systems suitable for various applications, from surveillance to disaster support missions. Hence, achieving high-precision trajectory tracking is crucial for their successful deployment. This research focuses on modelling, parameter identification, and Linear Quadratic Regulator (LQR) control design for quadrotors, aiming to enhance their trajectory following capabilities. The quadrotor dynamics are a sixth degree-of-freedom (6DOF) equation of motion derived from Newton’s second law, encompassing moment of inertia, centre of gravity, weight, and thrust of propeller parameters. Experimental measurements are conducted to accurately determine these parameters, ensuring a realistic representation of the quadrotor system. Subsequently, a linearized model is constructed to provide a suitable plant for control system development. The LQR control design is intended to improve the trajectory tracking performance. This control strategy is validated through simulation and practical experiments, demonstrating its effectiveness in achieving high-precision trajectory following capabilites. The proposed approach demonstrates that LQR control effectively guides the quadrotor to resemble a predefined trajectory, experiencing only 3 % overshoot observed during the initial phase of flight.

Design of infinite horizon LQR controller for discrete delay systems in satellite orbit control: A predictive controller and reduction method approach

In the realm of satellite orbit control, powerful controller design plays a pivotal role in minimizing fuel consumption and ensuring orbit stability. This article introduces an advanced approach to the design of a Linear Quadratic Regulator (LQR) controller with an infinite horizon, tailored for discrete delay systems. The proposed methodology integrates predictive control with a reduction method, aiming for optimality while addressing performance and system constraints. Formulating the control problem as a quadratic program, the predictive control method generates a sequence of control inputs using a reducing horizon strategy. Stability analysis, employing Lyapunov-Krasovsky functions and linear matrix inequalities, yields delay-independent conditions for exponential convergence. A numerical example showcases the controller's effectiveness in maintaining orbit and reducing fuel consumption, underlining its capacity to achieve control objectives despite uncertainties and time delays. This research contributes to robust control strategies in satellite orbit systems, enhancing control performance and operational efficiency.

LQR controller design for stabilization of non-linear DIP system based on ABC algorithm

Inverted pendulum systems, such as double or single, rotational or translational inverted pendulums are non-linear and unstable, which have been the most dominant approaches for control systems. The double inverted pendulum is one kind of a non-linear, unstable system, multivariable, and strong coupling with a wide range of control methods. To model these types of systems, many techniques have been proposed so that motivating researchers to come up with new innovative solutions. The Linear Quadratic Regulator (LQR) controller has been a common controller used in this field. Meanwhile, the Artificial Bee Colony (ABC) technique has become an alternative solution for employing Bee Swarm Intelligence algorithms. The research solutions of the artificial bee colony algorithm in the literature can be beneficial, however, the utilization of discovered sources of food is ineffective. Thus, in this paper, we aim to provide a double inverted pendulum system for stabilization by selecting linear quadratic regulator parameters using a bio-inspired optimization methodology of artificial bee colony and weight matrices Q and R. The results show that when the artificial bee colony algorithm is applied to a linear quadratic regulator controller, it gains the capacity to autonomously tune itself in an online process. To further demonstrate the efficiency and viability of the suggested methodology, simulations have been performed and compared to conventional linear quadratic regulator controllers. The obtained results demonstrate that employing artificial intelligence (AI) together with the proposed controller outperforms the conventional linear quadratic regulator controllers by more than 50 % in transient response and improved time response and stability performance

Fractional‐order filter‐based enhanced full state feedback control design with multi‐objective formulation using grey wolf optimization algorithm

AbstractA perturbed fractional‐order filter (FOF)‐based LQR control design with multiple performance indices is proposed in this paper. Three variants of the FOF‐based linear quadratic regulator (LQR) system have been proposed along with choices on the commensurate order of state feedback law. The multi‐objective problem formulation has been done with three performance objectives to address the different issues of design. The first performance index constitutes a weighted sum of ITAE (Integral Time of Absolute Error) and the difference between the eigenvalues of the plant and the LQR controlled plant, chosen to minimize the large oscillations as well as relative stability with respect to eigenvalues. A new lower bound of the singular values have been addressed to ensure the robust stability of the system through the formulation of a second performance objective. It is the minimization of singular values of the return ratio matrix at the plant's input. The third and the last performance index aims minimization of maximum singular values of the perturbation transfer function at the plant's output, so as to guarantee no closed‐loop right half plane zeros. The multi‐objective optimization problem is solved and the optimal solutions are obtained via Grey Wolf Optimization Algorithm. The simulation results validate the performance and effectiveness of the proposed control design.

A two-stage dimension-reduced dynamic reliability evaluation (TD-DRE) method for vibration control structures based on interval collocation and narrow bounds theories

Due to the uncertainties from modeling, manufacturing, and working environments, many vibration active control systems usually show dynamic uncertain properties. Hence structural reliability estimation benchmarking to full-cycle vibratory responses is vitally important. In this study, a novel two-stage dimension-reduced dynamic reliability evaluation (TD-DRE) method for linear quadratic regulator (LQR) controlled structures is developed. This method combines interval uncertainties and the time-variant reliability (TVR) concept. In the first stage, the Taylor series expansion is employed to analyze several typical limit states for definition of the time-discretized dynamic reliability. Then the interval collocation method tackles the solution. In the second stage, the TVR problem is indeed transformed to a time-invariant reliability (TIR) problem. Furthermore, the narrow bounds theorem deduces the presented TD-DRE index. Eventually, two application examples are utilized to verify the effectiveness and accuracy of the proposed method. The proposed TD-DRE is more accurate than the traditional first-order Taylor expansion and more effective than the first-passage reliability evaluation method. This method can provide a reference and an initial value for further design, and improve the efficiency of LQR controller design in practical engineering.

Model-free design of stochastic LQR controller from a primal–dual optimization perspective

To further understand the underlying mechanism of various reinforcement learning (RL) algorithms and also to better use the optimization theory to make further progress in RL, many researchers begin to revisit the linear–quadratic regulator (LQR) problem, whose setting is simple and yet captures the characteristics of RL. Inspired by this, this work is concerned with the model-free design of stochastic LQR controller for linear systems subject to Gaussian noises, from the perspective of primal–dual optimization. We first reformulate the stochastic LQR problem as a constrained non-convex optimization problem, which is shown to have strong duality. Then, to solve this non-convex optimization problem, we propose a model-based primal–dual (MB-PD) algorithm based on the properties of the resulting Karush–Kuhn–Tucker (KKT) conditions. We also give a model-free implementation for the MB-PD algorithm by solving a transformed dual feasibility condition. More importantly, we establish the connection between the proposed MB-PD algorithm and classical policy iteration algorithm, which provides a novel primal–dual optimization perspective to understand the common RL algorithms. Finally, we provide a high-dimensional case study to show the performance of the proposed algorithms.

An Improvement in LQR Controller Design based on Modified Chaotic Particle Swarm Optimization and Model Order Reduction

The diverse engineering and scientific applications are stated through complex and high-order systems. The significant difficulties of these systems are the complications of modeling, analyzing, and controlling. It is easier to examine simpler models for more physical insights than more complex models and result in lower-ordering controllers that are easier to implement. The model order reduction (MOR) was used to simplify the computational difficulty of such complications and was later developed intensively for use with increasingly CDS. In this paper, a new Modified Chaos Particle Swarm Optimization (MCPSO) technique is employed to get a reduced-order model of a large scale system and design a Linear Quadratic Regulator (LQR) based controller. The mod uses the combination of advantages of basic PSO algorithms and chaotic algorithms. It becomes an excellent algorithm with fast convergence, few control parameters, simple execution, and avoidance of local extremes. In addition to combining the chaotic algorithm, CPSO also improved the weight parameter w, adjusting it to the dynamic attenuation direction. First, efficient reduced-order model parameters are obtained for original higher-order systems based on the MCPSO. Then linear quadratic regulator (LQR (controller parameters optimized for the reduced-order model. The goodness of the proposed method is evaluated through a numerical example. The experimental results indicate that the proposed technique’s reduced order model provides an excellent close approximation to the original system.

State-Space Modeling and Active Vibration Control of Smart Flexible Cantilever Beam with the Use of Finite Element Method

The aim of this study is to design a Linear Quadratic Regulator (LQR) controller for the active vibration control of a smart flexible cantilever beam. The mathematical model of the smart beam was created on the basis of the Euler-Bernoulli beam theory and the piezoelectric theory. State-space and finite element models used in the LQR controller design were developed. In the finite element model of the smart beam containing piezoelectric sensors and actuators, the beam was divided into ten finite elements. Each element had two nodes and two degrees of freedom were defined for each node, transverse displacement, and rotation. Two Piezoelectric ceramic lead Zirconate Titanate (PZT) patches were affixed to the upper and lower surfaces of the beam element as pairs of sensors and actuators. The location of the piezoelectric sensor and actuator pair changed and they were consecutively placed on the fixed part, the middle part, and the free end of the beam. In each case, the design of the LQR controller was made considering the first three dominant vibratory modes of the beam. The effect of the position of the sensor-actuator pair on the beam on the vibration damping capability of the controller was investigated. The best damping performance was found when the sensor-actuator pair was placed at the fixed end.

System Identification and LQR Controller Design with Incomplete State Observation for Aircraft Trajectory Tracking

This paper presents a controller design process for an aircraft tracking problem when not all states are available. In the study, a nonlinear-transport aircraft simulation model was used and identified through Maximum Likelihood Principle and Extended Kalman Filter. The obtained mathematical model was used to design a Linear–Quadratic Regulator (LQR) with optimal weighting matrices when not all states are measured. The nonlinear aircraft simulation model with LQR controller tracking abilities were analyzed for multiple experiments with various noise levels. It was shown that the designed controller is robust and allows for accurate trajectory tracking. It was found that, in ideal atmospheric conditions, the tracking errors are small, even for unmeasured variables. In wind presence, the tracking errors were proportional to the wind velocity and acceptable for small and moderate disturbances. When turbulence was present in the experiment, state variable oscillations occurred that were proportional to the turbulence intensity and acceptable for small and moderate disturbances.

Singularly Perturbed Modeling and LQR Controller Design for a Fuel Cell System

Modeling and control of proton-exchange membrane fuel cells (PEMFC) has become a very popular research topic lately due to the increasing use of renewable energy. Despite this fact, most of the work in the current literature only studies standard dynamical models without taking into consideration possible parasitics such as small gas flow perturbations that could be available in the system. This paper addresses this issue by elaborating on time-scale modeling of an augmented eighteenth-order PEMFC-reformer system via singular perturbation theory. The latter captures time scales that arise in the model due to the presence of small perturbations. Specifically, a novel and efficient algorithm that helps identify the presence of different time-scales is developed. In addition, the method converts an implicit singularly perturbed model into an explicit equivalent where the time-scales are evident. Using this algorithm, a complete singularly perturbed dynamic model of the augmented eighteenth-order PEMFC-reformer system is obtained. Modeling of the PEMFC-reformer system is followed by linear quadratic regulator (LQR) design for the individual time-scales present in the system.

Nonprobabilistic Reliable LQR Design Method for Active Vibration Control of Structures with Uncertainties

The linear quadratic regulator (LQR), which allows a natural consideration of the constraints on state variables and control effort, is widely used in modern control theory for the active control of structural vibration. However, uncertainties are unavoidable in the design of active control systems. Thus, in the design process of an LQR controller, it is important to deal with these uncertainties, which are always bounded and have a great influence on the final performance of the closed-loop system. In this paper, the nonprobabilistic reliability concept is extended to the selection of weighting matrices for determining the LQR controller. First, a new uncertainty propagation analysis method for an active control system is presented. To adapt to the special problem of controller design, a novel nonprobabilistic reliability index is proposed. The reliability and safety margin of the closed-loop system can be continuously characterized by this modified nonprobabilistic reliability index. The weighing matrices used to design the LQR controller via the Riccati equation are finally determined by solving a reliability-based optimization problem. The proposed method preserves the excellent characteristics of LQR control theory and provides a quantitative method for selecting the weight matrices for LQR controller design in consideration of the inherent uncertainties. Two numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed method. The results from these examples reveal that the uncertainties in active control system design can be addressed from a reliability perspective.

Design and Performance Analysis of LQR Controller for Stabilizing Double Inverted Pendulum System

Double inverted pendulum (DIP) is a nonlinear, multivariable and unstable system. The inverted pendulum which continually moves toward an uncontrolled state represents a challenging control problem. The problem is to balance the pendulum vertically upward on a mobile platform that can move in only two directions (left or right) when it is offset from zero stat. The aim is to determine the control strategy that deliver better performance with respect to pendulum's angles and cart's position. A Linear-Quadratic-Regulator (LQR) technique for controlling the linearized system of double inverted pendulum model is presented. Simulation studies conducted in MATLAB environment show that the LQR controller are capable of controlling the multi output double inverted pendulum system. Also better performance results are obtained for controlling heavy driven part DIP system.

Active control of a non-linear landing gear system having oleo pneumatic shock absorber using robust linear quadratic regulator approach

This paper deals with the active control of a non-linear active landing gear system equipped with oleo pneumatic shock absorber. Runway induced vibration can cause reduction of pilot’s capability of control the aircraft and results the safety problem before take-off and after landing. Moreover, passenger–crew comfort is adversely affected by vertical vibrations of the fuselage. The active landing gears equipped with oleo pneumatic shock absorber are highly non-linear systems. In this study, uncertain polytopic state space representation is developed by modelling the pneumatic shock absorber dynamics as a mechanical system with non-linear stiffness and damping properties. Then, linear matrix inequalities-based robust linear quadratic regulator controller having pole location constraints is designed, since the classical linear quadratic regulator control design is dealing with linearized state space models without considering the non-linearities and uncertainties. Thereafter, numerical simulation studies are carried out to analyse aircraft response during taxiing. Bump- and random-type runway irregularities are used with various runway class and wide range of longitudinal speed. Simulation results revealed that neglecting the non-linear dynamics associated with oleo pneumatic shock absorber results significant performance degradation. Consequently, it is demonstrated that proposed robust linear quadratic regulator controller has a superior performance in terms of passenger–crew comfort and operational safety when compared to classical linear quadratic regulator.

Model Order Reduction and Linear Quadratic Regulator Controller Design on A Large Scale Linear Time Invariant System

Model Order Reduction and Linear Quadratic Regulator Controller Design on A Large Scale Linear Time Invariant System

Linear-quadratic regulator controller design for quadrotor based on pigeon-inspired optimization

Purpose The purpose of this paper is to propose a new algorithm for linear-quadratic regulator (LQR) controller of a quadrotor with fast and stable performance, which is based on pigeon-inspired optimization (PIO). Design/methodology/approach The controller is based on LQR. The determinate parameters are optimized by PIO, which is a newly proposed swarm intelligent algorithm inspired by the characteristics of homing pigeons. Findings The PIO-optimized LQR controller can obtain the optimized parameters and achieve stabilization in about 3 s. Practical implications The PIO-optimized LQR controller can be easily applied to the flight formation, autonomous aerial refueling (AAR) and detection of unmanned aerial vehicles, especially applied to (AAR) in this paper. Originality/value This research applies PIO to optimize the tuning parameters of LQR, which can considerably improve the fast and stabilizing performance of attitude control. The simulation results show the effectiveness of the proposed algorithm.

Tuning of LQR controller for an experimental inverted pendulum system based on The Bees Algorithm

Stabilizing of an inverted pendulum (IP) system is a main problem for researchers working on control theory. Balancing of an inverted pendulum system is one of the major benchmark problems in the control system community. This paper presents optimal tuning of linear quadratic regulator (LQR) controller with The Bees Algorithm (BA) for a linear inverted pendulum. In this paper, a metaheuristic approach which is a nature-inspired search method that mimics the foraging behavior of honey bees is used for design of LQR controller to obtain optimal performance. In LQR controller design, state (Q) and control (R) weighting matrices are basic parameters of LQR which are tuning by designer using trial and error method in usually. The Bees Algorithm optimizes the weighting matrices of the LQR controller so that it can move the cart to a desired position with the minimum change in pendulum’s angle from vertically upright position during the movement. The tuned LQR controller is benchmarked on the linear inverted pendulum experimental device (IP02) that is manufactured by QUANSER Company. After description of the system and The Bees Algorithm, the paper gives the experimental results obtained from the IP02 system to demonstrating the efficiency of the tuning of the LQR controller. Simulation and experimental results are given graphically to show the success of controller. As a result of the paper, the performance of LQR controller shows the effectiveness of The Bees Algorithm which is a diversity method for provide an efficient solution to conventional trial and error design approach.

Proportional-derivative linear quadratic regulator controller design for improved longitudinal motion control of unmanned aerial vehicles

This study investigates the longitudinal motion control of unmanned aerial vehicles through a simulation in MATLAB. The linear model of an unmanned aerial vehicle is applied to controllers to explicate the longitudinal motion of the unmanned aerial vehicle. The ideal performance for an unmanned aerial vehicle is to achieve the desired response instantly with 100% precision. However, currently available controllers need further improvements. Thus, a proportional-derivative linear quadratic regulator controller is developed and compared with a proportional-integral-derivative controller, a linear quadratic regulator controller, and a proportional linear quadratic regulator controller. The proportional-derivative linear quadratic regulator controller enhances the response of the system by reducing settling time by more than 95% compared with other available controllers. Additionally, the proportional-derivative linear quadratic regulator improves the root mean square error by almost 50% compared with the proportional linear quadratic regulator controller and improves rise time by almost 96% with reasonable overshoot.

LQR and MPC controller design and comparison for a stationary self-balancing bicycle robot with a reaction wheel

A self-balancing bicycle robot based on the concept of an inverted pendulum is an unstable and nonlinear system. To stabilize the system in this work, the following three main components are required, i. e., (1) an IMU sensor that detects the tilt angle of the bicycle robot, (2) a controller that is used to control motion of a reaction wheel, and (3) a reaction wheel that is employed to produce reactionary torque to balance the bicycle robot. In this paper, we propose three control strategies: linear quadratic regulator (LQR), linear model predictive control (LMPC), and nonlinear model predictive control (NMPC). Several simulation tests have been conducted in order to show that our proposed control laws can achieve stabilizaton and make the system balance. Furthermore, LMPC and NMPC controllers can deal with state and input constraints explicitly.

Quaternion-Based LQR Spacecraft Control Design Is a Robust Pole Assignment Design

AbstractRecently, a fully controllable quaternion spacecraft model was proposed. Based on this controllable quaternion model, an analytic linear-quadratic regulator (LQR) design for a spacecraft control system was obtained. Although the LQR design is based on the linearized quaternion model, it was shown that the design of the LQR control system globally stabilizes the original nonlinear spacecraft system. Because in spacecraft design practice engineers normally use models with simplified system dynamics and cost function (which has diagonal structure and excludes external torques) to make the design complexity manageable, this paper will show that the LQR control design is a robust pole assignment design; therefore, the controller is insensitive to the modeling error and external disturbances of nonlinear spacecraft systems. All these properties make this design very attractive. First, the designed controller locally achieves the desired performance because LQR design is optimal for the linearized system...