Linear Quadratic Regulator Theory Research Articles

LQR controller design for portable power supply based on flyback converter

Abstract The stability of the power repair equipment guarantees the reliability of the repair of electrical equipment. The following problems exist in the power supply of the emergency equipment: the load equipment randomly cuts in the power supply network; load equipment requires fast dynamic performance of the power supply. This challenges the stability of the portable power supply with the Flyback converter as the core. In this paper, an optimal feedback control strategy based on linear quadratic regulator (LQR) theory is proposed to improve the dynamic and steady-state performance of the Flyback converter. First, the state-averaged space model of the Flyback is derived and established. Second, an output voltage feedback integral controller is introduced to eliminate the steady-state error of the output voltage. Next, according to the LQR optimal control theory, the control model of the Flyback converter has been established, and the parameter design of the controller has been carried out by obtaining the optimal feedback gain matrix of the system. Finally, the simulation models are implemented with an output power of 120 W and a switching frequency of 50 kHz. The simulation results prove that the LQR controller provides superior performance than the traditional PI controller.

Saturated Trajectory Tracking Controller in the Body-Frame for Quadrotors

This paper introduces a quadrotor trajectory tracking controller comprising a steady-state optimal position controller with a normed input saturation and modular integrative action coupled with a backstepping attitude controller. First, the translational and rotational dynamical models are designed in the body-fixed frame to avoid external rotations and are partitioned into an underactuated position system and a quaternion-based attitude system. Secondly, a controller is designed separately for each subsystem, namely, (i) the position controller synthesis is derived from the Maximum Principle, Lyapunov, and linear quadratic regulator (LQR) theory, ensuring the global exponential stability and steady-state optimality of the controller within the linear region, and global asymptotic stability is guaranteed for the saturation region when coupled with any local exponential stable attitude controller, and (ii) the attitude system, with the quaternion angles and the angular velocity as the controlled variables, is designed in the error space through the backstepping technique, which renders the overall system, position, and attitude, with desirable closed-loop properties that are almost global. The overall stability of the system is achieved through the propagation of the position interconnection term to the attitude system. To enhance the robustness of the tracking system, integrative action is devised for both position and attitude, with emphasis on the modular approach for the integrative action on the position controller. The proposed method is experimentally validated on board an off-the-shelf quadrotor to assess the resulting performance.

Communication topology for AUV formation consensus based on optimal cost control under time-varying communication and input delays

This study investigates the optimal communication topology for consensus in autonomous underwater vehicle (AUV) formations, considering both leader-following and leaderless scenarios. Initially, a second-order dynamic model of the AUV is established using the feedback linearization method. To address time-varying delays, a universal predictor-based consensus protocol is introduced, capable of operation with or without a designated leader. Subsequently, two global quadratic cost functions are formulated based on distinct consensus errors. Utilizing linear quadratic regulator (LQR) theory and matrix theory, the optimal communication topologies and associated gain matrices are deduced. Specifically, for leader-following AUV formations, the optimal communication topology is identified as an unevenly weighted star-shaped directed spanning tree. Conversely, for leaderless formations, the optimal topology adopts a complete digraph, which indicates high interaction requirements. To accommodate practical constraints, a suboptimal topology is devised, featuring a directed spanning tree. These findings offer a theoretical framework for selecting communication topologies in AUV formations, with numerical examples provided to validate the proposed approaches.

Kalman state estimator for aircraft structural load alleviation using eigenpair assignment

Process of finding the “best estimate” from noisy signals amounts to “filtering out” the noise. Many methods exist to filter the unwanted noise. A tried and tested method is to use a Kalman Filter which not only cleans up the signals but can also be used to provide signal estimates, for use in reduced order feedback control. Typically, Kalman filter gains are computed using the Linear Quadratic Regulator theory. Reduced order feedback control has been applied in aircraft control problems. One such application area is in synthetic load alleviation, structural loads that arise due to gusts and or control surface deflections. Aircraft structural load alleviation necessitates the use of robust feedback control. The controllers are required to provide load alleviation in cases where the feedback signals are contaminated with noise or missing due to sensor failures. In this paper a method of computing the Kalman gains for the purposes of reduced order feedback is presented, which completely specifies the error dynamics of the estimator in terms of its eigenvalues and associated eigenvectors. The state estimator synthesized using the proposed approach has excellent noise rejection properties and shown to be robust and can be successfully used in reduced order state feedback control, specifically in Aircraft Structural Load Alleviation control schemes. The proposed scheme demonstrates reduction in structural loads.

Position and Attitude Tracking Controllers Using Lyapunov Transformations for Quadrotors

In this paper, a novel feedback control strategy for quadrotor trajectory tracking is designed and experimentally tested with proof of exponential stability, using the Lyapunov transformations theory. The controller is derived from an inner-outer loop control structure, namely by considering the position system coupled through an interconnection term with the attitude system. For the design of the position controller, the considered dynamics are worked on the body frame, which is uncommon in the literature, and its synthesis derives from theories such as Pontryagin’s maximum principle, Lyapunov theory, and Linear Quadratic Regulator (LQR), which ensure Input-to-state stability, steady-state optimality, and global exponential stability. The attitude system is based on an error quaternion parameterization via a nonlinear coordinate transformation matrix followed by a state input feedback, rendering the system linear and time-invariant. Under a correct transformation, LQR theory ensures almost exponential stability and steady-state optimality for the overall interconnected closed-loop systems. Experimental and simulation results illustrate the performance of the tracking system onboard a quadrotor.

Controller Design for Congestion Avoidance Based on Several Optimization Techniques

The world has become more like a small community thanks to the internet, which connects millions of people, businesses, and pieces of technology for a variety of uses. Because of the significant influence these networks have on our lives, maintaining their efficiency is important, which necessitates addressing issues like congestion. In this study, PI-controller gains are adjusted using a variety of optimization strategies to regulate the nonlinear TCPAQM model. This controller commits controlled pressured signaling characteristics and modifies computer network congestion. First manual tune PI-Controller are used; then several optimization techniques were used to tune PI-controller gains (Particle Swarm Optimization (PSO), Ant-Colony Optimization (ACO) and Simulated Annealing algorithm (SA)) and then Linear Quadratic Regulator theory are used. To test the reliability and effectiveness of each of the suggested controllers, several tests utilizing varied network parameter values, different queue sizes, and extra disturbances were conducted. MATLAB was used for all experiments., the results show the superiority of the LQR controller over PI controller with both manual and optimal tuning techniques.

CAV Traffic Control to Mitigate the Impact of Congestion from Bottlenecks: A Linear Quadratic Regulator Approach and Microsimulation Study

This work investigates traffic control via controlled connected and automated vehicles (CAVs) using novel controllers derived from the linear-quadratic regulator (LQR) theory. CAV-platoons are modeled as moving bottlenecks impacting the surrounding traffic with their speeds as control inputs. An iterative controller algorithm based on the LQR theory is proposed along with a variant that allows for penalizing abrupt changes in platoon speeds. The controllers use the Lighthill-Whitham-Richards (LWR) model implemented using an extended cell transmission model (CTM) which considers the capacity drop phenomenon for a realistic representation of traffic in congestion. The impact of various parameters of the proposed controller on the control performance is analyzed. The effectiveness of the proposed traffic control algorithms is tested using a traffic control example and compared with existing proportional-integral (PI) and model predictive control (MPC) controllers from the literature. A case study using the TransModeler traffic microsimulation software is conducted to test the usability of the proposed controller as well as existing controllers in a realistic setting and derive qualitative insights. It is observed that the proposed controller works well in both settings to mitigate the impact of the jam caused by a fixed bottleneck. The computation time required by the controller is also small making it suitable for real-time control.

An integration planning and control method of intelligent vehicles based on the iterative linear quadratic regulator

This paper proposes an integrated strategy of planning and control to address the poor quality of the generated trajectory by intelligent vehicles. The trajectory planner is designed based on the iterative linear quadratic regulator (ILQR) theory, taking into account the nonlinear dynamics of the vehicle in the trajectory planning layer. The planner directly outputs the control commands that act on the actuator, achieving the integration of planning and control. To improve the control performance of the trajectory planner under extreme operating conditions, a stability controller is developed using the sliding mode control method (SMC). Additionally, a torque distributor is designed to coordinate the control requirements of the trajectory planner and the stability controller, with the optimization goal of minimizing the tire loading rate. Finally, complex scenario tests are carried out under various working conditions to evaluate the performance of the integrated strategy. The results indicate that the integrated strategy outperforms the decoupled strategy in terms of control performance and exhibits better adaptability across various working conditions and scenarios. Even if the reference trajectory is unreasonable, the vehicle can still perform smooth, efficient, and comfortable obstacle avoidance and lane-changing maneuvers. By working in tandem, the stability controller and the trajectory planner can expand the planning and control boundaries of the system, thereby significantly enhancing the system's ability to respond to extreme working conditions.

Multivariable-Controlled Shunt Compensator for Damping Subsynchronous Interactions in Electrical Distribution Networks

In recent years, the massive integration of converter-interfaced devices to electrical distribution networks has lead to appearance of poorly-damped subsynchronous interactions in the grid. These oscillation modes amplify electrical disturbances originated from the grid and may cause an interaction between grid elements threatening the system stability. Shunt compensators at the distribution level have already been used to improve the damping of these networks. However, their control system strongly depends on the distribution network dynamics, that is difficult to know at all times. In this paper, a multivariable controller for a shunt compensator is proposed. First, the dynamics of a distribution network with high penetration of electronic interfaces is explored in order to understand the interactions. Then, the controller of the compensator is designed based on an estimated model of the plant so that no previous information of the network is required. This controller is based on a state-feedback controller and a state observer, both designed by using the linear quadratic regulator (LQR) theory. Analytical results and computer simulations are used to verify the effectiveness of the controller. The main findings are validated in a laboratory environment comprising two 15 kW distributed generators, a 15 kW compensator and a 75 kW grid emulator.

Backstepping based model reference adaptive control for nuclear reactor with matched and unmatched uncertainties

In this work, a Backstepping Model Reference Adaptive Controller (BMRAC) is designed to control the total power of a Pressurized Water Reactor (PWR). It ensures robustness against matched and unmatched parameter uncertainties and disturbances. Reactor core dynamics based on normalized point kinetics equations with six delayed neutron groups is considered, while parametric variations in thermal feedback coefficients, control rod worth and heat transfer coefficients are considered as uncertainties. Reactivity disturbance is considered as unmatched disturbance and external disturbances that enter from control input are considered as matched disturbance. A stable reference model is designed using linear quadratic regulator (LQR) theory to generate setpoints for reactor power control. The disturbances and uncertainties are estimated by comparing reactor dynamics with a stable predictor model. Matched parameter uncertainties and disturbances are compensated with robust MRAC. To compensate for unmatched disturbance, an integrator backstepping is designed. Projection-based adaptive laws are used to provide robustness to the designed controller. The ultimate boundedness of all signals is verified through Lyapunov analysis. Efficiency of the proposed controller is demonstrated through simulation studies, controller performance is compared with conventional LQR controller and adaptive MRAC controller.

Delay-Dependent Optimal Load Frequency Control for Sampling Systems With Demand Response

The continuous-time model is widely used to design load frequency control (LFC) schemes for power systems with delays and demand response (DR). In practice, however, the LFC system works in a sampled-data manner. The plant operates continuously while the control updates every few seconds. This paper focuses on the sampled-data LFC system with DR and the delay. Based on the linear quadratic regulator (LQR) theory, an optimal control scheme is designed to regulate the DR and generation units in a sampled-data manner. The LQR controller can be applied to the LFC system with constant and time-varying delays, regardless of DR. An observer is also designed to make the state-feedback control law feasible. The observer is efficient in that its gain is delay independent. Case studies on the one-area and multi-area LFC systems are conducted to verify the effectiveness of the proposed LFC controller. Simulation results also demonstrate the excellent stability and performance of the proposed LQR controller in the sampled-data LFC-DR system.

Sensor and Process Noises Reduction using a Luenberger State Estimator with a Stability Augmentation System for a Hypersonic Transport Aircraft

This paper firstly, presents an autopilot strategy for a Hypersonic Transport Aircraft (HST) using a Stability Augmentation System (SAS) with a Luenberger estimator. The SAS is designed using Linear Quadratic Regulator (LQR) theory which, for HST, benefits the guaranteed robust dynamic stability provided three theoretical requirements are met. The Luenberger estimator is incorporated into the autopilot design to estimate the state variables of the aircraft for the SAS. In the dynamic response simulation, sensor and process noises are inserted into the mathematical model. However, to date, knowledge of the sensor and process noises at the speeds and heights where the aircraft will be flying is limited. The simulation shows that the Luenberger estimator significantly filters the noise. This is an advantage for the HST as prior knowledge of the noises is not necessary when designing the Luenberger estimator.

대잠전용 무인수상로봇의 자율운항시스템 설계를 위한 LQR 기반 경로추종 알고리즘 연구

This paper presents the usage of LQR-based intelligent path tracking methods to control the autonomous navigation system of an Unmanned Surface Robot (USR) in sea-clutter environmental fields. Since considerable interest in the Autonomous Navigation System (ANS) of USR in sea warfare has increased, the stability of the Autonomous Navigation System (ANS) control becomes of the utmost importance. When the USR moves in sea environments, the control stability of autonomous navigation systems becomes a major concern. The Path Tracking Controller (PTC) of USRs is one of the high potential subsystems that can be further improved to achieve more accurate, robust, and comfortable tracking performance. The stability of the PTC should be taken into consideration while performing the path tracking task in autonomous navigation. To solve the automatic waypoint tracking stability of independently-developed PTCs, a waypoint tracking controller is designed based on the Linear Quadratic Regulator (LQR) theory with the kinematic model of USRs. Then, a simulation analysis of the designed waypoint tracking controller is performed, and the results are compared at two different speeds. Additionally, complex waypoint tracking tests under realistic sea conditions are performed based on the designed waypoint tracking controller. Results show good tracking performance, which indicates that the designed waypoint tracking controller can adapt to high-speed USRs.

Speed-Sensorless Control of Induction Machines with LC Filter for Geothermal Electric Submersible Pumping Systems

A speed-sensorless state-feedback controller for induction machines (IMs) with LC filter is proposed. The speed and state estimation is based on a speed-adaptive observer, requiring only the measurement of the filter input currents. The motor currents are controlled by a state-feedback controller with prefilter and integral control action, in order to achieve fast and asymptotic set point tracking. Observer and controller gains are calculated offline using linear quadratic regulator (LQR) theory and updated online (gain-scheduling) in order to attain stability and improve controller performance in the whole operation range. Implementation aspects, such as discretization of the control system and reduction of computational effort, are taken into account as well. The proposed control scheme is validated by simulations and experimental results, even for critical operating conditions such as speed zero-crossings. It is shown that the overall control system performs very well under various load- and speed conditions; while its tuning remains simple which makes it attractive for industrial application such as geothermal electric submersible pumping (ESP) systems.

Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System

The present study uses linear quadratic regulator (LQR) theory to control a vibratory system modeled by a fractional-order differential equation. First, as an example of such a vibratory system, a viscoelastically damped structure is selected. Second, a fractional-order LQR is designed for a system in which fractional-order differential terms are contained in the equation of motion. An iteration-based method for solving the algebraic Riccati equation is proposed in order to obtain the feedback gains for the fractional-order LQR. Third, a fractional-order state observer is constructed in order to estimate the states originating from the fractional-order derivative term. Fourth, numerical simulations are presented using a numerical calculation method corresponding to a fractional-order state equation. Finally, the numerical simulation results demonstrate that the fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.

Path-Following Control Method for Surface Ships Based on a New Guidance Algorithm

A new type of path-following method has been developed to steer marine surface vehicles along desired paths. Path-following is achieved by a new hyperbolic guidance law for straight-line paths and a backstepping control law for curved paths. An optimal controller has been improved for heading control, based on linear quadratic regulator (LQR) theory with nonlinear feedback control techniques. The control algorithm performance is validated by simulation and comparison against the requirements of International Standard IEC62065. Deviations are within the allowable range of the standard. In addition, the experimental results show that the proposed method has higher control accuracy.

A nonsmooth Newton method for the design of state feedback stabilizers under structure constraints

The paper proposes a method for structured state-feedback controllers design for linear time-invariant systems. A necessary and sufficient condition for structured state-feedback stabilizability of linear systems, making an appeal to the linear-quadratic (LQ) regulator theory, is first proposed. The latter is presented in the form of a nonlinear matrix equation. Then, it is recast as a nonsmooth unconstrained equation using projection onto the positive semi-definite matrices cone. Thereby, a nonsmooth Newton’s iterative algorithm, based on the Clarke generalized Jacobian of said projection, is proposed. This method has a guaranteed local convergence. Finally, numerical examples illustrate the effectiveness of the proposed method.

Discrete-Time DC-Link Voltage and Current Control of a Grid-Connected Inverter with LCL-Filter and Very Small DC-Link Capacitance

The paper presents a controller design for grid-connected inverters (GCI) with very small dc-link capacitance that are coupled to the grid via an LCL filter. The usual controller designs would fail and result in instability. The proposed controller has a cascaded structure with a current controller as inner control loop and an outer dc-link voltage controller. The controller design is performed in discrete time and it is based on a detailed stability analysis of the dc-link voltage controller to determine the controller parameters which guarantee stability for all operating points. The inner loop is a state-feedback current controller that is designed based on the discrete linear-quadratic regulator (DLQR) theory. An additional integral error feedback assures steady-state accuracy of the current control loop. The simulation and experimental results validate performance and stability of proposed controller design.

Vibration Control using Fractional Derivative Feedback

This study aims at applying linear quadratic regulator (LQR) theory to control a vibratory system modelled by a fractional-order differential equation. First, an iteration-based method for solving the algebraic Riccati equation is proposed to obtain a fractional-order LQR. Next, Newmark-𝛽method is modified to be able to simulate a fractional-order differential equation. Finally, as an example, a vibration control problem is considered for a generalized Voigt model. Numerical calculations demonstrate that the Fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.

Feedback control of chaotic systems using multiple shooting shadowing and application to Kuramoto-Sivashinsky equation.

We propose an iterative method to evaluate the feedback control kernel of a chaotic system directly from the system's attractor. Such kernels are currently computed using standard linear optimal control theory, known as linear quadratic regulator theory. This is however applicable only to linear systems, which are obtained by linearizing the system governing equations around a target state. In the present paper, we employ the preconditioned multiple shooting shadowing (PMSS) algorithm to compute the kernel directly from the nonlinear dynamics, thereby bypassing the linear approximation. Using the adjoint version of the PMSS algorithm, we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto-Sivashinsky equation. We compare the computed kernel with that produced by the standard linear quadratic regulator algorithm and note similarities and differences. Both kernels are stabilizing, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.