Classical Linear Quadratic Regulator Research Articles

Parabolic set simulation for reachability analysis of linear time-invariant systems with integral quadratic constraint

This paper describes the computation of reachable sets and tubes for linear time-invariant systems with an unknown input bounded by integral quadratic constraints, modeling e.g. delay, rate limiter, or energy bounds. We define a family of paraboloidal overapproximations. These paraboloids are supported by the reachable tube on touching trajectories. Parameters of each paraboloid are expressed as a solution to an initial value problem. Compared to previous methods based on the classical linear quadratic regulator, our approach can be applied to unstable systems as well. We tested our approach on large scale systems.

Modified LQR Technique for Fuel-Cell-Integrated Boost Converter

Fuel cells (FCs) are an alternative source of energy whose by-products are heat, vapor, and water. For medium-power and standalone operation, an FC is integrated with a boost converter, as the required voltage level is higher than the supply. Control of such a converter with a slow dynamic source requires a fast-acting controller, since the demand executed by the controller affects the action taken by the hydrogen valve in FCs. In this article, a linear-quadratic regulator (LQR) is developed to control the power converter. In practical applications, the conventional LQR produces offset in output voltage due to effective series resistance (ESR) of the output capacitor. To make the LQR dynamic with load perturbation, information of inductor reference current is required, which changes with variation in load current. To solve these two problems, two modifications in the classical LQR have been proposed to minimize the impact of ESR (m-LQR) and incorporate control action based on the reference inductor current (M-LQR). The proposed methods are compared with the conventional average current-mode control technique. The M-LQR-based scheme is found to outperform the classical m-LQR technique in terms of improved transient dynamics and reduced offset at the steady state. In addition to numerical simulations, the proposed control techniques have been validated experimentally.

Swarm Optimized Simple Adaptive Controller for Spacecraft Proximity Operations Trajectory Tracking

Abstract Adaptive control design allows for the management of systems with time varying or unknown dynamics. Despite their versatility, few well defned design techniques exist for some classes of adaptive controller. Without analytical techniques it is difcult to prove the efcacy of an adaptive controller design. One solution to this issue is the application of parametric search techniques to adaptive controller design. This paper explores the application of diferential evolution on the simple adaptive control law formulation and compares its solution to one found using particle swarm optimization. Afterwards, variations on these techniques, namely the selection particle swarm optimization and self-adaptive diferential evolution, are implemented and their results compared. The final swarm-optimized controller is compared to a classical Linear Quadratic Regulator (LQR) controller, and a manually designed simple adaptive controller for precision trajectory tracking control of spacecraft proximity operations. Parametric search techniques are able to determine controller parameters that produce a superior control response. Swarm-optimization techniques determine controllers with parameters drastically different from manually designed efforts.

Optimal Control Synthesis of Semi Active Vehicle Suspension

The paper discusses the problems related with the synthesis of a linear quadratic (LQ) regulator of a semi active vehicle suspension. The synthesis combines the control through output variables with such by the input excitation i.e. classical LQ regulator with Compensator. The physical nonlinearities of the controllable magneto-rheological semi-active damper are taken into an account with the including in the feedback control circuit an inverse damper model. . A reduced suspension model is used to illustrate the synthesis (quarter car model).

A multi‐time‐delay compensation controller using a Takagi–Sugeno fuzzy neural network method for high‐rise buildings with an active mass damper/driver system

SummaryAn active mass damper/driver (AMD) control system with a single mass has such problems as the excessive weight of the auxiliary mass and the insufficient capacity of its driving equipment. It is necessary to work through multiple subsystems to achieve effective control of high‐rise buildings. However, the time‐delay effect in each subsystem impedes its application in engineering practices. In the paper, an augmented system based on a zero‐order hold is proposed for discrete‐time systems with multiple time delays, and then the system is designed according to the compensation strategy using a classical linear quadratic regulator algorithm. After that, the sample data obtained from the zero‐order hold compensation controller is trained through a Takagi–Sugeno fuzzy neural network method. Finally, a new simplified compensation controller is designed to further shorten the time consuming calculation on the premise of guaranteeing its control effects and parameters. To verify its effectiveness, an AMD system in a high‐rise building is regarded as an example, and the proposed methodology is also applied to an experiment of a four‐story frame. Both results demonstrate that the method can enhance the performance of an AMD system with multiple time delays.

USING GENETIC ALGORITHM IN OPTIMIZING THE MATRIX OF LQR CONTROLLER FOR NONLINEAR INVERTED PENDULUM SYSTEM

In this article, the author used the matlab software to simulate and then compared the results between the classical LQR (Linear Quadratic Regulator) controller and another method to adjust the matrix parameters toward optimization of the LQR controller. It is the GA (Genetic Algorithm) method to optimize the matrix of the LQR controller, and the results have been verified on the nonlinear pendulum model. The Genetic Algorithm is a modern control algorithm, which is widely applied in research and practice. The main objective of this article is to use the GA algorithm in order to optimize the matrix parameters of LQR controller, whichcontrolled the position and angle of the nonlinear inverted pendulum at the stable balance point. The matlab-based simulating results showed that the system has operated properly to the requirements and the output response has reached an equilibrium position of about 2.5 seconds.

Gain scheduled integral linear quadratic control for quadcopter

The findings of this paper are focused on the dynamics and control of a quadcopter using a modified version of a Linear Quadratic Regulator (LQR) control approach. The classical LQR control approach is extended to include an integral term to improve the quad copter tracking performance. The mathematical model is derived using the Newton-Euler method for the nonlinear six DOF model that includes the aerodynamics and detailed gyroscopic moments as a part of the system identification process. The linearized model is obtained and it is characterized by the heading angle (yaw angle) of the quadcopter. The adopted control approach is utilizing the LQR method to track several trajectories i.e. helical and lissajous curve with significant variation in the yaw angle. The integral term is introduced to the controller in order to minimize the steady state errors observed. The controller is modified to overcome difficulties related to the continuous changes in the operation points and to eliminate the chattering that was observed in the control technique. Numerical non-linear simulations are performed using MATLAB & Simulink to illustrate to accuracy and effectiveness of the proposed controller.  Â

Adaptive control of earthquake-excited nonlinear structures with real-time tracking on prescribed performance criteria

A model reference adaptive control method is proposed to suppress the nonlinear responses of earthquake-excited structures with the proposed implementation of performance tracking criteria. The model reference concept is adopted to propose a comprehensive framework for real-time tracking on prescribed performances, and a deterministic reference model is presented to explicitly represent the prior-designed structural performances. The tracking error and the time-varying adaptive control law are systematically presented and designed by using the backstepping theory and the Lyapunov stability analysis. Little prior or explicit knowledge of structural nonlinear behavior is considered for the proposed adaptive backstepping law. Two performance schemes, inspired by the classical linear quadratic regulator control theory and the unknown disturbance cancelation, are developed. Illustrative examples of a bilinear single-degree-of-freedom system and a seismically base-isolated benchmark building structure are employed with a detail illustration and discussion about the control performance of the proposed methodology. Finally, an additional parametric study on the issues of the actuator saturation, the optimum parameter selection, and the switching delay is presented and discussed.

Observer-based H2 controller design for a vibration isolation stage having hybrid electromagnets

Vibration isolation systems based on hybrid electromagnets, consisting of electromagnet and permanent magnet, have a potential usage in many industrial areas, such as clean room design, transportation, semiconductor manufacturing, suspension systems, and robotic surgery due to providing mechanical contact free vibration isolation. Using permanent magnets in the electromagnet structure has some crucial advantages, such as a minimized volume and a more compact structure. Furthermore, the essential force for levitation of vibration isolation stage can be generated by only the permanent magnet(s), which means, by using hybrid electromagnets, magnetic levitation can be achieved with considerably low energy consumption against possible vibrations. This property is called zero-power behavior. However, the main problems of magnetic levitation process are as follows: it has highly nonlinear nature even if it can be linearized; it has unstable pole(s), which makes the system vulnerable in terms of stability. In recent years, linear matrix inequality-based design of controllers has received considerable attention and become very popular due to their ability to satisfy multiobjective design requirements. However, an observer-based H2 controller design for a vibration isolation system having hybrid electromagnets has not been considered yet. Therefore, the linear matrix inequality-based controller is employed to minimize the effect of disturbances on the following objectives, such as vibration isolation, zero-power property, and protection of the levitation gap. The effectiveness of the proposed method is shown with the numerical simulation studies and compared with classical Linear Quadratic Regulator (LQR) approach.

Utility of probability density evolution method for experimental reliability-based active vibration control

The utility of probability density evolution method for reliability-based active vibration control of a cantilevered flexible beam is experimentally investigated. In this respect, an optimal linear quadratic regulator (LQR) is utilized together with an observer to design an online full-state feedback controller. In order to design a well-performing controller and to simulate the controller performance, a system model is obtained via identification techniques. Reliability tests are consequently performed to verify the effectiveness of the presented reliability assessment method as a foundation for reliability-based control. Subsequently, a hybrid metaheuristic optimization scheme of continuous ant colony system is used for simultaneous integrated tuning of the LQR and observer parameters through all processes. Optimization of the controller and the observer is performed with the objective of system structural reliability. An optimized version of classical LQR controller is also developed for making comparisons between the two control methods. Finally, a comparison is made between the mean power consumption of the 2 controllers. It is realized that the reliability-based controller decreases operating costs without compromising system safety.

Neural Network Based Vibration Control of Seismically Excited Civil Structures

This study proposes a neural network based vibration control system designed to attenuate structural vibrations induced by an earthquake. Classical feedback control algorithms are susceptible to parameter changes. For structures with uncertain parameters they can even cause instability problems. The proposed neural network based control system can identify the structural properties of the system and avoids the above mentioned problems. In the present study it is assumed that a full state of the structure is known, which means the at each floor horizontal displacements and rotations about the vertical axis are measured. Additionally, it is assumed the acceleration signal coming from the earthquake is also available. The proposed neural control strategy is compared with the classical linear quadratic regulator (LQR) not only in terms of displacement responses, but also required control forces. Moreover, the influence of different weighting matrices on performance of the proposed control strategy has been presented.The effectiveness of the neuro-controller has been demonstrated on two numerical examples: a simple single degree of freedom (DOF) structure and a multi-DOF structure representing a twelve story building. Both structures under consideration have been excited with El Centro acceleration signal. The results of numerical simulations on the SDOF system indicate that using neuro-controller it would be possible to obtain smaller amplitudes as compared with the LQ regulator, but it would require higher control effort.

Piezoelectric Actuator Placement Optimization and Active Vibration Control of a Membrane Structure

The placement optimization of piezoelectric actuators and active vibration control of a membrane structure are studied in this paper. The classical linear quadratic regulator controllers are designed to suppress the unwanted vibration. Simulation results indicate that the optimal locations of piezoelectric actuators are affected deeply by the additional mass and stiffness of actuators, the computational efficiency of particle swarm optimizer is higher than that of genetic algorithm for this particular problem, and the control performance of optimally placed actuators is better than that of non-optimally placed actuators.

Stable Optimal Control and Semicontractive Dynamic Programming

We consider discrete-time infinite horizon deterministic optimal control problems with nonnegative cost per stage, and a destination that is cost free and absorbing. The classical linear-quadratic regulator problem is a special case. Our assumptions are very general, and allow the possibility that the optimal policy may not be stabilizing the system, e.g., may not reach the destination either asymptotically or in a finite number of steps. We introduce a new unifying notion of stable feedback policy, based on perturbation of the cost per stage, which in addition to implying convergence of the generated states to the destination, quantifies the speed of convergence. We consider the properties of two distinct cost functions: $J^*$, the overall optimal, and $\hat J$, the restricted optimal over just the stable policies. Different classes of stable policies (with different speeds of convergence) may yield different values of $\hat J$. We show that for any class of stable policies, $\hat J$ is a solution of Bel...

Dynamic model and active vibration control of a membrane antenna structure

This paper studies a dynamic model and active vibration control of a membrane antenna structure. Based on the finite element method (FEM), the dynamic model of the membrane antenna structure is established. Piezoelectric actuators are used to suppress the vibration of the structure and their optimal locations on the membrane are determined using the optimization method, where an efficient numerical criterion depended on controllability Grammian is used as optimization criterion and the particle swarm optimizer (PSO) is used as optimization algorithm. Active controllers are designed by the classical linear quadratic regulator (LQR) method. Simulation results indicate that the vibration modes and dynamic responses obtained by the dynamic model established in this paper coincide well with the results of the software ABAQUS; vibration of the structure can be suppressed effectively by the piezoelectric actuators, and optimal placed actuators not only can produce better control effectiveness but also need smaller control cost.

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.

Seismic retrofit of frame structures using passive systems based on optimal control

This paper proposes a new seismic modification methodology for optimal design of multistory frame structures to avoid damage and attain a robust response to extreme hazards. The methodology determines added damping devices of optimal size at strategic locations and modifies story stiffnesses, by solving a newly formulated constrained optimization problem. The objective is to minimize a cost function representing total energy while satisfying the equations of motion and allowable interstory drifts for given severe ground motions. The proposed methodology combines the classical linear quadratic regulator, which searches for an optimal gain matrix, with an analysis-redesign iterative procedure to produce a robust analysis redesign (RAR) iterative approach. A new algorithm approximates the equivalent stiffness to fit the displacement gains. At convergence, the RAR procedure will yield optimal sizing and new topology for the designed system. The RAR procedure is developed here for shear structures. However, RAR can be utilized for complex structures by modeling them first as equivalent shear-type structures having stiffness properties that yield the same maximum interstory drifts for the given records. Two numerical examples of shear-type structures are examined. These examples use structures already having an “optimal” configuration, which are further optimized using the methodology developed herein. In addition, a numerical example comprised of a 10-story moment resisting frame structure is presented to illustrate the use of the equivalent shear structure approach. The results of the exemplified retrofitted moment resisting frame structure show great improvements in its dynamic response.

Online Adaptive Optimal Control of Vehicle Active Suspension Systems Using Single-Network Approximate Dynamic Programming

In view of the performance requirements (e.g., ride comfort, road holding, and suspension space limitation) for vehicle suspension systems, this paper proposes an adaptive optimal control method for quarter-car active suspension system by using the approximate dynamic programming approach (ADP). Online optimal control law is obtained by using a single adaptive critic NN to approximate the solution of the Hamilton-Jacobi-Bellman (HJB) equation. Stability of the closed-loop system is proved by Lyapunov theory. Compared with the classic linear quadratic regulator (LQR) approach, the proposed ADP-based adaptive optimal control method demonstrates improved performance in the presence of parametric uncertainties (e.g., sprung mass) and unknown road displacement. Numerical simulation results of a sedan suspension system are presented to verify the effectiveness of the proposed control strategy.

Structured State Feedback Controllers and Optimal Submanifold Stabilization

We study optimal stabilization of submanifolds. We show that the adjoint equation of the Hamiltonian formalism associated with such optimal control problems has structured solutions. As a consequence, optimal controls for submanifold stabilization are necessarily structured state feedbacks. In particular, we show that optimal controllers depend linearly on a function mapping to the normal spaces of the submanifold. We connect our findings to the classical linear quadratic regulator via classes of Riccati differential equations and algebraic Riccati equations. We illustrate our findings on the synchronization problem, on the broadcasting problem, and on the pattern generator problem.

Attitude coordination of multiple spacecraft for space debris surveillance

This paper discusses the attitude coordination of a formation of multiple spacecraft for space debris surveillance. Off-the-shelf optical sensors and reaction wheels, with limited field of view and control torque, respectively, are considered to be used onboard the spacecraft for performing the required attitude maneuvers to detect and track space debris. The sequence of attitude commands are planned by a proposed algorithm, which allows for a dynamic team formation, as well as performing suitable maneuvers to eventually point towards the same debris. A control scheme based on the nonlinear state dependent Riccati equation is designed and applied to the space debris surveillance mission scenario, and its performance is compared with those of the classic linear quadratic regulator and quaternion feedback proportional derivative controllers. The viability and performance of the coordination algorithm and the controllers are validated through numerical simulations.

Linear-quadratic regulator. I. a new solution

In this paper, the classical linear-quadratic regulator problem is solved via use of the linear matrix inequality technique. This approach is shown to yield the optimal solution obtained by using the matrix Riccati equation. Various undesirable effects are discussed, which may appear when applying other solution methods known from the literature; numerical examples are presented.