Linear Quadratic Regulator (LQR)

The PID controller, which has proportional, integral and derivative elements, is widely applied in feedback control of industrial processes. These controllers are described with their simple structure and principle. PID controllers also provide good performance for various systems. However, PID method in many cases such as parameter variations or disturbances is not appropriate. In order to overcome some problems that faced by PID controller, the other type of control methods can be developed such as Linear-Quadratic Regulator (LQR) optimal control. LQR is a control scheme that gives the best possible performance with respect to some given measure of performance. The performance measure is a quadratic function composed of state vector and control input.

This paper purpose the procedure to estimate conity (λ), longitudinal creep coefficients (f11) also lateral creep coefficients (f22) [5]. The Continuous- Time Recursive Least Absolute Error (C-T RLAE) with Variable Forgetting Factor (VFF) method to estimate solid axle wheelset parameter in Linear Quadratic Regulator (LQR) of two axle railway are presented. The Continuous- Time Recursive Least Absolute Error (C-T RLAE) with Variable Forgetting Factor (VFF) are used to overcome this bias problem. The Continuous Time Recursive Least Absolute Error (C-T RLAE) are designed for Linear time invariant (LTI) system but the Variable Forgetting Factor (VFF) are used for time varying system. The performance of Continuous Time Recursive Least Absolute Error (C-T RLAE) originally and Continuous- Time Recursive Least Absolute Error (C-T RLAE) with Variable Forgetting Factor (VFF) methods are compared. Linear Quadratic Regulator (LQR) was applying to stabilise wheelset traveling especially at high speeds wheelset parameters. Continuous- Time Recursive Least Absolute Error (C-T RLAE) with Variable Forgetting Factor (VFF) methods give good estimated of railway wheelset compared Continuous- Time Recursive Least Absolute Error (C-T RLAE). From this combination from (C-T RLAE) with (VFF) can reduce estimation bias.

Abstract— This paper proposes the emulation of different photovoltaic (PV) materials and technologies using an innovative PV array emulator based on Linear Quadratic Regulator (LQR), this electronic power device aims to reproduce faithfully the real PV module behavior independently on environmental conditions change, it allows scientists, industrials and researchers to carry out their measurements and experiences on PV systems without depending on PV panels, which require the sun to perform tests and do not allow repetitive measurements at the desired temperature (T) and irradiance (G). Moreover, PV modules actually are very expensive and require a large area to reach some powers, all these limitations and others are handled using this designed PV emulator. Simulation results using Matlab Simulink software are given and analyzed in order to evaluate the performances of the developed equipment and to judge its efficiency and capacity to track rapidly and accurately the I-V characteristic of different PV modules.

Non-linear characteristic of tire forces is the main cause of vehicle lateral dynamics instability, while direct yaw moment control is an effective method to recover the vehicle stability. In this paper, an optimal linear quadratic regulator (LQR) controller for roll-yaw dynamics to articulated heavy vehicles is developed. For this purpose, the equations of motion obtained by the MATLAB software are coded and then a control law is introduced by minimizing the local differences between the predicted and the desired responses. The influence of some parameters such as the anti roll bar, change the parameters of the suspension system and track wide in articulated heavy vehicles stability has been studied. The simulation results show that the vehicle stability can be remarkably improved when the optimal linear controller is applied

This paper deals with the study and comparison of passive and active landing gear system of the aircraft and dynamic responses due to runway irregularities while the aircraft is taxying. The dynamic load and vibration caused by the unevenness of runway will result in airframe fatigue, discomfort of passengers and the reduction of the pilot’s ability to control the aircraft. One of the objectives of this paper is to obtain a mathematical model for the passive and active landing gears for full aircraft model. The main purpose of current paper is to design linear quadratic regulator (LQR) for active landing gear system that chooses damping and stiffness performance of suspension system as control object. Sometimes conventional feedback controller may not perform well because of the variation in process dynamics due to nonlinear actuator in active control system, change in environmental conditions and variation in the character of the disturbances. To overcome the above problem, we have designed a controller for a second order system based on Linear Quadratic Regulator. The performance of active system is compared with the passive landing gear system by numerical simulation. The results of current paper in compared with the previous work mentioned in reference, demonstrates 37.04% improvement in body acceleration, 20% in fuselage displacement and 13.8% in the shock strut travel. The active landing gear system is able to increase the ride comfort and good track holding by reducing the fuselage acceleration and displacement and load induced to airframe caused by runway excitation.

Abstract: Today’s aircraft consist of number of automatic controller that helps the airplane crew in airplane management and piloting the aircraft. Usually, the control strategies of aircraft can be grouped into two categories as follows: lateral and longitudinal control. In longitudinal control, the pitch angle has been employed for an aircraft system. Pitch of an aircraft may be described as a rotation around the lateral axis. An elevator is used to control pitch of aircraft which is located at the back of an airplane. So in this paper, a pitch control of an aircraft has been implemented by using Fuzzy controller. The stability of aircraft system is of major concern in real interface system. Thus, aircraft system must be accurate having stable response of aircraft system and should approaches to reference over a certain interval of time i.e. it should not oscillate over a long time. So this work exhibits the design of fuzzy control strategy for controlling the pitch angle of an aircraft. Two control strategies: Linear quadratic regulator (LQR) control and Fuzzy control strategies have been employed and compared. After that a comparative analysis of these control strategies has been done. The whole system has been implemented in MATLAB SIMULINK environment.

Where Q is symmetric positive semi-definite state weighting matrix of order × and R is symmetric positive definite control weighting matrix of order × The choice of the element Q and R allows the relative weighting of individual state variables and individual control inputs as well as relative weighting state vector and control vector against each other. The weighting matrices Q and R are important components of an LQR optimization process. The compositions of Q and R elements have great influences of system performance. The designer is free to choose the matrices Q and R, but the selection of matrices Q and R is normally based on an iterative procedure using experience and physical understanding of the problems involved. Commonly, a trial and error method has been used to construct the matrices Q and R elements. This method is very simple and very familiar in linear quadratic regulator application. However, it takes long time to choose the best values for matrices Q and R. The number of matrices Q and R elements are dependent on the number of state variable (n) and the number of input variable (m), respectively. The diagonal-off elements of these matrices are zero for simplicity. If diagonal matrices are selected, the quadratic performance index is simply a weighted\ integral of the squared error of the states and inputs. The term in the brackets in equation (9) above are called quadratic forms and are quite common in matrix algebra. Also, the performance index will always be a scalar quantity, whatever the size of Q and R matrices .The conventional linear quadratic regulator problem is to find the optimal control input law u* that minimizes the performance index under the constraints of Q and R matrices. The closed loop optimal control law is defined as:

Line impedance is also responsible for the voltage distortion caused by the circulation of non-sinusoidal current. It degrades the performance of the power conditioners due to its effects on the corresponding control and synchronization systems [3]-[4]. A new preliminary modelling methodology was introduced where a more realistic model applied for grid connected systems is presented while considering the dynamics of the distorted point of common coupling (PCC) voltages. This approach was shown to be an improved alternative using linear quadratic regulator (LQR) controllers and the Kalman filter (KF) algorithm [5] applied to these systems. The KF is a recursive algorithm that is well known for dealing with dynamic systems corrupted by uncertainties or noise and which has been widely studied and used in very different applications [6]. In addition to the controller, the reference generation plays an important role to achieve the compensation objectives [7]–[9]. It is assumed that the frequency of the grid is constant, and only harmonics are detected. In a recent paper [10], a mathematical model that is capable of identifying harmonics and unbalance is proposed while using a KF with less computational effort. The method is based on single phase filter. However, this method still considers a constant frequency and the model is derived by rotating axes, which leads to a time- variant model even under constant frequency, in contrast with the time invariant model used in this work. In the time variant model, it would also need an extra block for proper synchronization that is not the case in this proposal.

are a function of the longitudinal velocity to optimally tune the gains for wider range of speeds. Therefore, the controller belongs to the class of gain scheduling Robust Linear Quadratic Regulator (RLQR). It is noted that, the proposed RLQR method also allows the decoupled design of the LQR control action and the robust action, thus avoiding the use of time-consuming tuning for the selection of the LQ weights. The LQ weights can be chosen without considering model imperfects and disturbances. Then, based on the Riccati solution, the robust term can be designed to suppress uncertainties. The closed-loop tracking error dynamics are analytically proven to be globally uniformly ultimately bounded and an upper-bound for the ultimate bound (i.e., the maximum residual error when time goes to infinity [22]) is formulated, where, the proof is carried out by considering the plant dynamics as a parameter-varying system [23]. Hence, unwanted dynamics, which can be induced by gain scheduling strategies [24], cannot emerge. In addition, the ultimate bound is inversely proportional to the gain of the robust action, confirming the advantage provided by the proposed feedback control action. For the numerical validation, the novel RLQR is embedded in IPG CarMaker in which a prototype electric Range Rover Evoque is modelled. A quantitative comparison clearly shows that the novel RLQR outperforms gain-scheduling LQR control solutions proposed by [13], in terms of residual tracking error, peak yaw rate error and absolute value of the control action. Finally, experimental results on the same prototype electric Range Rover Evoque with individually controlled motors on front and rear axles confirm the applicability and the effectiveness of the proposed control strategy to real scenarios. Furthermore, an additional detailed comparative analysis, carried out with experimental data, points out that superior close-loop tracking performance can be achieved when the RLQR replaces gain-scheduling LQRs available in the literature [13].

Alhamdulillah, thanks to Allah SWT, with His willing for giving me the opportunity to complete this Final Year Project which is title Active Control of Quarter-Car Suspension System using Linear Quadratic Regulator. This final year report was prepared for Faculty of Electrical Engineering, University Teknikal Malaysia Melaka (UTeM), basically for student in final year to complete the undergraduate program that leads to the degree of Bachelor of Engineering in Electrical.

In this paper modeling of an inverted pendulum is done using Euler – Lagrange energy equation for stabilization of the pendulum [10]. The controller gain is evaluated through state feedback and Linear Quadratic optimal regulator controller techniques and also the results for both the controller are compared. The SFB controller is designed by Pole- Placement technique. An advantage of Quadratic Control method over the pole- placement techniques is that the former provides a systematic way of computing the state feedback control gain matrix. LQR controller is designed by the selection on choosing. The proposed system extends classical inverted pendulum by incorporating two moving masses. The motion of two masses that slide along the horizontal plane is controllable.

Because of its extensive application, the control of a nonlinear system that is uncertain is an area of study that has been very actively researched. There are efforts to apply artificial intelligent to a controller to enhance the damping of low frequency oscillations in a SMIB. This paper aims to highlight the implementation of a few prevalent optimal control methods like linear Gaussian regulator (LGR), linear quadratic regulator (LQR), H controller design, H2 controller design on the system and evaluation of the performance of each control method for use in the PSS.

Where Q is symmetric positive semi-definite state weighting matrix of order and R is symmetric positive definite control weighting matrix of order The choice of the element Q and R allows the relative weighting of individual state variables and individual control inputs as well as relative weighting state vector and control vector against each other. The weighting matrices Q and R are important components of an LQR optimization process. The compositions of Q and R elements have great influences of system performance. The designer is free to choose the matrices Q and R, but the selection of matrices Q and R is normally based on an iterative procedure using experience and physical understanding of the problems involved. Commonly, a trial and error method has been used to construct the matrices Q and R elements. This method is very simple and very familiar in linear quadratic regulator application. However, it takes long time to choose the best values for matrices Q and R. The number of matrices Q and R elements are dependent on the number of state variable (n) and the number of input variable (m), respectively.

Abstract— this paper presented comparison of the time specification performance between two type of controller for a Double Inverted Pendulum system. Double Inverted Pendulum is a non-linear ,unstable and fast reaction system. DIP is stable when its two pendulums allocated in vertically position and have no oscillation and movement and also inserting force should be zero. The objective is to determine the control strategy that to delivers better performance with respect to pendulum angle’s and cart position. In this paper simple multi PD controller designed on the theory of pole placement and its performance is compared with Linear Quadratic Regulator controller using MATLAB and Simulink.

In this paper, an optimal control scheme for wind turbine output torque and power regulation under the influence of wind disturbances is presented. The system considered is a dynamic mechanical-based model with pitch and ge- nerator torque actuators for controlling the pitch and generator torque. The performance of linear matrix inequality (LMI) formalism of linear quadratic regulator (LQR); linear quadratic regulator with integral action (LQRI) and model predictive control (MPC) were compared in response to a step change in wind disturbance. It is shown by Matlab simulation that the LQRI outper- formed both LQR and MPC controllers.

The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear-quadratic regulator (LQR). The LQR is an important part of the solution to the LQG problem. Like the LQR problem itself, the LQG problem is one of the most fundamental problems in control theory,[9].

Figure 12 Figure 13 Design and performance of ACLCD by WLQR algorithm Here, a multiresolution based wavelet controller WLQR, originally proposed by Basu and Nagarajaiah[38], is designed [r]

The attitude control problem of the kinetic kill vehicle is studied in this work. A new mathematical model of the kinetic kill vehicle is proposed, the linear quadratic regulator technique is used to design the optimal atti- tude controller, and the pulse-width pulse-frequency modulator is used to shape the continuous control command to pulse or on-off signals to meet the requirements of the reaction thrusters. The methods to select the appropriate parameters of pulse-width pulse-frequency are presented in detail. Numerical simulations show that the performance of the LQR/PWPF approach can achieve good control performance such as pseudo-linear operation, high accuracy, and fast enough tracking speed.

This study focused on traffic regulation of metro systems. A real-time delay recovery problem for metro loop lines using constrained Model Predictive Controller (MPC) and constrained Linear Quadratic Regulator (LQR) controllers in the presence of operational constraints on control action is considered. These two types of controllers were implemented to metro traffic model for recovering delays. The performance of the MPC and LQR were compared for traffic regulation. Simulation results show that constrained MPC performed better in comparison with the constrained LQR. Whereas, the MPC was computationally intensive, especially for large- scale metro traffic systems.

The quadrotor unmanned aerial vehicle is a great platform for control systems research as its nonlinear nature and under-actuated configuration make it ideal to synthesize and analyze con- trol algorithms. After a brief explanation of the system, several algorithms have been analyzed in- cluding their advantages and disadvantages: PID, Linear Quadratic Regulator (LQR), Sliding mode, Backstepping, Feedback linearization, Adaptive, Robust, Optimal, L 1 , H ∞ , Fuzzy logic and Artificial