Optimised output feedback PID controller for LQR inability

The main objective of this research is to design a state-feedback controller for the rotary inverted pendulum module utilizing the linear quadratic regulator (LQR) technique. The controller maintains the pendulum in the inverted (upright) position and is robust enough to reject external disturbance to maintain its stability. The research work involves three major contributions: mathematical modeling, simulation, and real-time implementation. To design a controller, mathematical modeling has been done by employing the Newton-Euler, Lagrange method. The resulting model was nonlinear so linearization was required, which has been done around a working point. For the estimation of the controller parameters, MATLAB LQR function has been utilized. Simulation has been performed for the designed controller and it also has been implemented and tested over the real inverted pendulum. From the results, it is vivid that the designed controller keeps the inverted pendulum in an upright position and rejects the disturbances and falling under gravitational force by adjusting the rotation of the horizontal link.

Rotary inverted pendulum (RIP) system is a nonlinear, non-minimum phase, unstable and underactuated system. Controlling such system can be a challenge and is considered a benchmark in control theory problem. Prior to designing a controller, equations that represent the behaviour of the RIP system must be developed as accurately as possible without compromising the complexity of the equations. Through Takagi-Sugeno (T-S) fuzzy modeling technique, the nonlinear system model is then transformed into several local linear time-invariant models which are then blended together to reproduce, or approximate, the nonlinear system model within local region. A parallel distributed compensation (PDC) based fuzzy controller using linear quadratic regulator (LQR) technique is designed to control the RIP system. The results show that the designed controller able to balance the RIP system.

In this paper, a particle swarm optimization (PSO) based linear-quadratic (LQ) state-feedback regulator is investigated. The parameters of LQ regulators are determined by PSO method. A practical example of a rotating inverse pendulum is provided to demonstrate the effectiveness of the PSO-based LQ regulators.

Friction stir welding (FSW) is a promising welding technology and is being extensively researched because it can produce high-quality joints. In the welding process, due to improper clamping, deformation of back support and other process variations, the axial force can vary significantly and produce welding defects. To solve this problem, a feedback controller of FSW axial force is desirable. However, time-delay, which widely exists within the FSW systems, may degrade the controller effect and make the system unstable. To regulate the axial force and compensate the time delay, this paper designs an axial force controller for FSW system with delay. At first, by utilizing a time delay compensation scheme, FSW systems with delayed control actions is transformed into standard differential equations without time delay terms. Then, an axial force controller is designed based on the linear-quadratic regulator (LQR) technique. Since the time delay is incorporated in the system model throughout the whole controller design process, the system performance and stability are assured and the control system is also simplified. Utilizing the LQR technique, aspects of system dynamic properties and simplicity are considered simultaneously. Experimental validations were carried out on a FSW platform. The experimental results demonstrate that the time delay is compensated effectively. The controller maintains the constant axial force and shows desirable dynamic behavior, even when the disturbance is encountered during the welding process.

The electric motor is a device that converts electrical power into mechanical power. The DC motor has some advantages compared to the AC motor on its easier way to control the speed or position and wide adjustable range. However, it still has some issues, such as uncertainty and disturbance; most existing controllers still provide bad system performance or have major drawbacks in solving these issues. The alternative solution for a control method with simple, easy to design, and can be implemented in a multi-input multi-output system is the integral state feedback control technique, such as in Linear Quadratic Regulator (LQR). The weakness of LQR control is that there is no exact method to determine the best value of matrix Q and R used to calculate the state feedback gain; trial or manual tuning method is mostly used. Therefore, this paper proposed an improved LQR control using Particle Swarm Optimization (PSO) method and the Kalman filter estimator. The PSO was used to find optimal Q and R-value, while the Kalman filter reduced the number of sensors used to measure the system state. Both performance and energy consumption were compared using the Integral of Absolute Error (IAE) and total energy consumption. The simulation results showed that the proposed method is superior in performance and energy compared to manually-tuned LQR. However, based on the hardware implementation, although the manually-tuned LQR was found to have the smallest energy consumption, the proposed method has the smallest IAE, which can reduce the IAE by 11.28% with only 1% higher energy compared to manually-tuned LQR.

This paper proposes the use Fuzzy logic controller and a Particle Swarm Optimization (PSO) based Linear Quadratic Regulator (LQR) controller for the balance of an electric unicycle. It also examines dynamics of the electric unicycle's schematic design, which has a backing seat for the passenger and is analogous to the Segway device equipped with a handle shaft for direction. A proportional derivative controller incorporated with a Fuzzy logic controller was designed for the suggested model of the electric unicycle for speed and position control (Fuzzy controller), and a Proportional Derivative (PD) controller was utilized to control the angle rate and angle of the unicycle. Comparison of the Fuzzy logic controller with a Particle Swarm Optimization (PSO) Linear Quadratic Regulator (LQR) controller was carried out based on error deviation and time response simulations. Command tracking or regulation of some initial state condition were examined and demonstrated by time based simulations of the modelled system. The results evidently indicate that the Fuzzy Proportional Derivative (PD) controller can effectively attain attitude equilibrium for the two wheeled unicycle and preventing it from falling down.

The use of active vibration control technique, though typically more expensive and difficult to implement compared to the traditionally used passive control approaches, is an effective way to control or suppress dangerous vibrations in structures caused by resonance or flutter due to the external disturbances, such as winds, waves, moving weights of human bodies, and earthquakes. From control perspective, it is not only sufficient to suppress the resonant modes of vibration but also to guarantee the overall stability of the structural system. In addition, the no-spillover property of the remaining large number of frequencies and the corresponding mode shapes must be maintained. To this end, the state-space based linear quadratic regulator (LQR) and \(H_{\infty }\) control techniques and physical-space based robust and minimum norm quadratic partial eigenvalue assignment (RQPEVA and MNQPEVA) techniques have been developed in the recent years. In contrast with the LQR and \(H_{\infty }\) control techniques, the RQPEVA and MNQPEVA techniques work exclusively on the second-order model obtained by applying the finite element method on the vibrating structure, and therefore can computationally exploit the nice properties, such as symmetry, sparsity, bandness, positive definiteness, etc., inherited by the system matrices associated with the second-order model. Furthermore, the RQPEVA and MNQPEVA techniques guarantee the no-spillover property using a solid mathematical theory. The MNQPEVA technique was originally developed for the deterministic models only, but since then, further extension to this technique has been made to rigorously account for parametric uncertainty. The stochastic MNQPEVA technique is capable of efficiently estimating the probability of failure with high accuracy to ensure the desired resilience level of the designed system. In this work, first a brief review of the LQR, \(H_{\infty }\), and MNQPEVA techniques is presented emphasizing on their relative computational advantages and drawbacks, followed by a comparative study with respect to computational efficiency, effectiveness, and applicability of the three control techniques on practical structures. Finally, the methodology of the stochastic MNQPEVA technique is presented and illustrated in this work using a numerical example with a representative real-life structure and recorded data from a California earthquake.

Parrot Mambo mini-drone is a readily available commercial quadrotor platform to understand and analyze the behavior of a quadrotor both in indoor and outdoor applications. This study evaluates the performance of three alternative controllers on a Parrot Mambo mini-drone in an interior environment, including Proportional–Integral–Derivative (PID), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC). To investigate the controllers’ performance, initially, the MATLAB®/Simulink™ environment was considered as the simulation platform. The successful simulation results finally led to the implementation of the controllers in real-time in the Parrot Mambo mini-drone. Here, MPC surpasses PID and LQR in ensuring the system’s stability and robustness in simulation and real-time experiment results. Thus, this work makes a contribution by introducing the impact of MPC on this quadrotor platform, such as system stability and robustness, and showing its efficacy over PID and LQR. All three controllers demonstrate similar tracking performance in simulations and experiments. In steady state, the maximal pitch deviation for the PID controller is 0.075 rad, for the LQR, it is 0.025 rad, and for the MPC, it is 0.04 rad. The maximum pitch deviation for the PID-based controller is 0.3 rad after the take-off impulse, 0.06 rad for the LQR, and 0.17 rad for the MPC.

This study investigates energy efficient control system for a small scale rocket recovery, utilizing thrust vector control (TVC) and reaction gas control (RGC) approaches. The dynamics of the rocket is managed through control input thruster forces and gimbaled angles, with the output parameters expressed in terms of velocity. Leveraging this model, several controllers: the linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), sliding mode control (SMC), and proportional integral derivative (PID) controllers were designed and implemented. A linearized version of the nonlinear dynamic model of the rocket, obtained through Jacobian linearization, facilitates the application of LQR and LQG controllers. For precise pitch and yaw control, the architecture incorporates a single-nozzle TVC actuation and four-sided gas thrusters. Simulation studies, conducted in MATLAB/Simulink, evaluated the energy consumption profiles of each control strategy during rocket recovery, with the PID controller used as a baseline for comparison. Without disturbances, the LQG controller reduced energy consumption by 17.63% compared to PID, while SMC and LQR achieved reductions of 14.65% and 14.24%, respectively. Under disturbed conditions, LQG, SMC, and LQR reduced energy consumption by 30.39%, 27.38%, and 27.30%, respectively. These results highlight the potential of advanced control techniques to optimize energy efficiency. Thus, this study provides valuable insights for the development of more sustainable and cost-effective reusable rocket systems by emphasizing energy-efficient recovery control.

In the present paper different approaches for flight control stabilization tuning of a prototype of Unmanned Aerial Vehicle (UAV) are investigated. The mathematical model of a multirotor airframe is presented and its dynamical system is described by means of Newton-Euler equations for a rigid body. The proposed controller systems for the stabilization of the altitude and the attitude of the multirotor around its hovering configuration are based on Proportional Derivative (PD) regulator tuned by optimization techniques such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) or, for the linearized system, by means of Linear Quadratic Regulator (LQR). In this case, PSO and GA are applied to set entries of LQR scheme. Control flight simulation results have been compared in terms of minimization of work carried out by the control actions. Performed simulations pointed out that all approaches lead to zero the error of the position along gravity acceleration direction, stop the rotation of UAV around body axes and stabilize the multirotor. The direct PSO tuning of the PD parameters (PD-PSO) shows better results in terms of errors, performance and speed of convergence.

In this paper, the performance of Model-Free Adaptive Control (MFAC) has been investigated on a novel and specific moving-mass controlled (MMC) flying robot system. The novel one-degree-of-freedom (1 DOF) MMC flying robot test bed presented in this paper has highly nonlinear and slow dynamics with a variable center of gravity (CoG) and moment of inertia. This makes the control of this system a challenging problem. One of the solutions to this challenge is the use of data-driven control methods, in particular, MFAC. This controller uses a data-driven model to control the system using only input and output (I/O) data. This paper compares this data-driven controller with proportional-integral-derivative (PID) control, and Linear Quadratic Regulator (LQR) as two model-free and model-based controllers which are widely used controllers in industry. The results of the comparison show that in the various scenarios applied, MFAC has a clear superiority over the PID and LQR, and its adaptive structure gives more freedom of action in the implementation of different scenarios and the presented noise. The results are obtained using the Integral Time Absolute Error (ITAE) criteria and the mean maximum error has also been compared in a Monte Carlo analysis. For a more detailed study, the amount of control energy consumption was also compared, which showed a clear superiority of the MFAC. Also, the robustness of the controller was demonstrated by introducing uncertainty in the plant parameters and by running 100 Monte Carlo simulations with random initial conditions. Finally, despite the PID controller, the MFAC followed the desired scenarios well and compared to LQR consumed less energy. The results demonstrate that the MFAC outperformed the PID and LQR controllers in the presence of random initial conditions and noise in terms of mean maximum error (70.4%)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(70.4\\%)$$\\end{document}, mean ITAE (91%)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(91\\%)$$\\end{document}, and energy consumption (46%)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(46\\%)$$\\end{document}.

This paper presents the designing and simulation for vertical moving control of Unmanned Air Vehicles (UAV) system using Proportional Integral Derivative (PID) control method, Linear Quadratic Regulator (LQR) control method and fuzzy logic control technique.The UAV system pose a challenger control problem that in it precise model of system is non-linear. The conventional PID controller, the LQR controller, Fuzzy Logic Controller (FLC) and Self-Tuning Fuzzy PID controller are designned based on the dynamic modeling of the system is derived from a suitable mathematical model to describe the vertical motion of a UAV. Simulation results in Matlab and Simulink show that by using self-tuning Fuzzy PID, the performance of vertical moving control system is improved significantly compared to Other controllers. DOI: http://dx.doi.org/10.11591/ijece.v3i5.3440

A simulated study of three different control strategies to stabilize an inverted pendulum on a cart is presented in this paper. Linear Quadratic Regulator (LQR), Integer order PID (IOPID) and Fractional order PID (FOPID) are designed, applied and compared. A nonlinear model of the cart-pendulum system is first derived and used later in the simulation of the control strategies aforementioned. The tuning parameters of all three controllers are obtained using Particle Swarm Optimization (PSO) technique, using the Integral of the Squared Error (ISE) index as fitting function, when a sequence of step reference inputs is applied. The simulation results, when applying a sequence of step references, show a better performance for the LQR, when using the cost function of the LQR as performance index. However, in the more general case of a sinusoidal references, the FOPID is the one that exhibits the best behavior.

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

Selecting appropriate weighting matrices for desired linear quadratic regulator (LQR) controller design using evolutionary algorithms is presented in this paper. Obviously, it is not easy to determine the appropriate weighting matrices for an optimal control system and a suitable systematic method is not presented for this goal. In other words, there is no direct relationship between weighting matrices and control system characteristics, and selecting these matrices is done by using trial and error based on designer’s experience. In this paper, we use the particle swarm optimization (PSO) method which is inspired by the social behavior of fish and birds in finding food sources to determine these matrices. Stable convergence characteristics and high calculation speed are the advantages of the proposed method. Simulation results demonstrate that in comparison with genetic algorithms (GAs), the PSO method is very efficient and robust in designing of optimal LQR controller. Key words: Linear quadratic regulator (LQR), weighting matrices, particle swarm optimization, genetic algorithm.