Linear Quadratic Regulator Approach Research Articles

A Lyapunov-based three-axis attitude intelligent control approach for unmanned aerial vehicle

A novel Lyapunov-based three-axis attitude intelligent control approach via allocation scheme is considered in the proposed research to deal with kinematics and dynamics regarding the unmanned aerial vehicle systems. There is a consensus among experts of this field that the new outcomes in the present complicated systems modeling and control are highly appreciated with respect to state-of-the-art. The control scheme presented here is organized in line with a new integration of the linear-nonlinear control approaches, as long as the angular velocities in the three axes of the system are accurately dealt with in the inner closed loop control. And the corresponding rotation angles are dealt with in the outer closed loop control. It should be noted that the linear control in the present outer loop is first designed through proportional based linear quadratic regulator (PD based LQR) approach under optimum coefficients, while the nonlinear control in the corresponding inner loop is then realized through Lyapunov-based approach in the presence of uncertainties and disturbances. In order to complete the inner closed loop control, there is a pulse-width pulse-frequency (PWPF) modulator to be able to handle on-off thrusters. Furthermore, the number of these on-off thrusters may be increased with respect to the investigated control efforts to provide the overall accurate performance of the system, where the control allocation scheme is realized in the proposed strategy. It may be shown that the dynamics and kinematics of the unmanned aerial vehicle systems have to be investigated through the quaternion matrix and its corresponding vector to avoid presenting singularity of the results. At the end, the investigated outcomes are presented in comparison with a number of potential benchmarks to verify the approach performance.

Multi‐input and multi‐output proportional‐integral‐derivative controller design via linear quadratic regulator‐linear matrix inequality approach

This study considers the problem of designing a multi-input and multi-output (MIMO) proportional-integral-derivative (PID) controller via direct optimal or suboptimal linear quadratic regulator (LQR) approach. To design the controller, first the MIMO PID design problem is transformed into a state feedback control and then the gains of the state feedback controller are chosen through an optimal or suboptimal LQR design. Given a minimal state space representation (A, B, C) of the plant, a necessary and sufficient condition (based on matrices A, C) for which the optimal problem (i.e. PID design via optimal LQR) is solvable is obtained. When this optimal problem is not solvable, a suboptimal solution (i.e. PID design via suboptimal LQR), if exists, is obtained by converting the problem into trace minimisation one, which is solved using linear matrix inequality-based method. Suitable examples are considered to illustrate the approaches.

Multimode Modified Positive Position Feedback to Control a Collocated Structure

Multimode modified positive position feedback (MMPPF) is proposed to suppress vibrations at multi-resonance frequencies in flexible collocated structures. Typically, flexible structures have large numbers of active modes in low frequency bandwidths, which make them susceptible to multi-frequency resonant vibrations. Hence, it is essential for the controller to have effective suppression on all participating modes. The MMPPF controller consists of a first- and a second-order compensator for each mode, as they are all set parallel for all active modes. Because of suppression performance sensitivity to controller gain parameters, proper gain selection is essential. Here, the linear quadratic regulator (LQR) approach and a proposed method called M-norm are used for gain optimization of the MMPPF controller. The optimized controller is then evaluated experimentally using a cantilever beam, enhanced by a piezoelectric actuator. According to the obtained results, the MMPPF controller reduces vibration amplitudes to the expected lower level, under both LQR and M-norm optimization methods. In some cases, vibration amplitude at the place of piezo-actuator is reduced to even less than the vibration amplitude level of the disturbance input at clamped end.

Optimal Placement of Piezoelectric Macro Fiber Composite Patches on Composite Plates for Vibration Suppression

This work presents a new methodology for the parametric optimization of piezoelectric actuators installed in laminated composite structures, with the objective of controlling structural vibrations. Problem formulation is the optimum location of a Macro Fiber Composite (MFC) actuator patch by means the maximization of the controllability index. The control strategy is based on a Linear Quadratic Regulator (LQR) approach. For the structural analysis, the modeling of the interaction between the MFC and the structure is made taking into account the active material as one of the orthotropic laminate shell layers. The actuation itself is modeled as an initial strain arising from the application of an electric potential which deforms the rest of the structure. Thereby, modeling the electric field and the electromechanical coupling within the actuator is avoided because these effects are considered analytically. Numerical simulations show that the structural model presents good agreement with numerical and experimental results. Furthermore, the results show that optimizing the location of the actuator in the structure helps the control algorithm to reduce induced structural vibration.

Towards the identification of heat induction in chip removing processes via an optimal control approach

This paper presents a linear-quadratic regulator (LQR) approach for solving inverse heat conduction problems (IHCPs) arising in production processes like chip removing or drilling. The inaccessibility of the processed area does not allow the measuring of the induced temperature. Hence the reconstruction of the heat source based on given measurements at accessible regions becomes necessary. Therefore, a short insight into the standard treatment of an IHCP and the related LQR design is provided. The main challenge in applying LQR control to the IHCP is to solve the differential Riccati equation. Here, a model order reduction approach is used in order to reduce the system dimension. The numerical results will show the accuracy of the approach for a problem based on data given by practical measurements.

Linear and nonlinear control strategies for formation and station keeping of spacecrafts within the context of the three body problem

The problem of spacecraft formation control and reconfiguration for halo orbit around the second libration point (L2) of the Sun–Earth Three Body (TB) system is investigated. Station keeping, reconfiguration and precision formation control of spacecrafts on halo orbits are performed via the use of the nonlinear Integral Sliding Mode (ISM) method as well as the optimal closed loop Linear Quadratic Regulator (LQR) approach. In this regard the nonlinear relative dynamics of deputy–chief spacecrafts are derived within the concept of the three body problem. The behavior of the two controllers are compared for different tasks of the formation mission in order to determine the more preferred strategy that fulfills the desired response behavior. In addition the controllers are evaluated for robustness in terms of their compensation against the major source of environmental disturbance, namely the Moon's gravity perturbation.

Decoupling trajectory tracking for gliding reentry vehicles

A decoupling trajectory tracking method for gliding reentry vehicles is presented to improve the reliability of the guidance system. Function relations between state variables and control variables are analyzed. To reduce the coupling between control channels, the multiple-input multiple-output (MIMO) tracking system is separated into a series of two single-input single-output (SISO) subsystems. Tracking laws for both velocity and altitude are designed based on the sliding mode control (SMC). The decoupling approach is verified by the Monte Carlo simulations, and compared with the linear quadratic regulator (LQR) approach in some specific conditions. Simulation results indicate that the decoupling approach owns a fast convergence speed and a strong anti-interference ability in the trajectory tracking.

Riccati-based Boundary Feedback Stabilization of Incompressible Navier--Stokes Flows

In this article a boundary feedback stabilization approach for incompressible Navier--Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. The nested iteration resul...

Digital sliding mode controller design for multiple time-delay continuous-time transfer function matrices with a long input–output delay

This paper extends the dominant eigenvector-based sliding mode control (SMC) design methodology, which was originally developed for delay-free continuous-time processes with known parameters, to the case of multiple time-delay continuous-time processes with known/unknown parameters. In addition, this paper presents a new prediction-based Chebyshev quadrature digital redesign methodology for indirect design of the digital counterpart of the analog sliding mode controller (ASMC) for multiple time-delay continuous-time transfer function matrices with either a long input delay or a long output delay. An approximated discrete-time model and its corresponding continuous-time model are constructed for multiple time-delay continuous-time stable/unstable dynamical processes with known/unknown parameters, using first the conventional observer/Kalman filter identification (OKID) method. Then, an optimal ASMC is developed using the linear quadratic regulator (LQR) approach, in which the corresponding sliding surface is designed using the user-specified eigenvectors and the scalar sign function. For digital implementation of the proposed non-augmented low-dimensional ASMC, a digital counterpart is designed based on the existing prediction-based digital redesign method and the newly developed prediction-based Chebyshev quadrature digital redesign method. Finally, a non-augmented low dimensional digital observer with a long input or output dead time is constructed for the implementation of the digitally redesigned sliding mode controller, to improve the performances of multiple time-delay dynamical processes. The effectiveness of the proposed method has been verified by means of two illustrative examples.

MBPOA-based LQR controller and its application to the double-parallel inverted pendulum system

As the performance of Linear Quadratic Regulator (LQR) controllers greatly depends on its weighting matrices, i.e. Q and R, designing these two matrices is one of the most important components in the LQR problem which is a tedious and challenging work in the applications of LQR. Hence, a novel LQR approach based on the Pareto-based Multi-objective Binary Probability Optimization Algorithm (MBPOA) is proposed in this paper, in which MBPOA is utilized to search for the optimal weighting matrices to relieve the effort of parameter settings and improve the control performance according to the pre-defined objective functions. By combining LQR with MBPOA, the optimal controllers can be obtained easily and effortless. Moreover, the control performance can be adjusted further conveniently to meet the requirements of applications as a set of Pareto-optimal LQR controllers is offered. The simulation and experiment results on the double-parallel inverted pendulum system demonstrate the effectiveness and efficiency of the developed MBPOA-based LQR method. Considering the characteristics such as robustness, the optimal dynamic performance and easy implementation without prior knowledge, the MBPOA-based LQR is a promising control approach for engineering applications.

Convex optimization of nonlinear feedback controllers via occupation measures

The construction of feedback control laws for underactuated nonlinear robotic systems with input saturation limits is crucial for dynamic robotic tasks such as walking, running, or flying. Existing techniques for feedback control design are either restricted to linear systems, rely on discretizations of the state space, or require solving a nonconvex optimization problem that requires feasible initialization. This paper presents a method for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). In contrast to traditional approaches based on Lyapunov’s criteria for stability, we rely on the notion of occupation measures to pose this problem as an infinite-dimensional linear program which can then be approximated in finite dimension via semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS which is well suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on six nonlinear systems. Comparisons to an infinite-horizon linear quadratic regulator approach and an approach relying on Lyapunov’s criteria for stability are also included in order to illustrate the improved performance of the presented technique.

Adaptive steering control of an electric unicycle

This paper presents techniques for adaptive steering control of an electric unicycle with a steering turning mechanism, in order to achieve self-balancing and speed tracking simultaneously. A state feedback control method using the linear quadratic regulator approach is first presented to simultaneously achieve self-balancing and speed tracking, and then an adaptive friction compensating term is proposed to overcome frictions caused by different terrain surfaces. The performance and merit of the proposed control method are well exemplified by conducting two simulations and five experiments on a laboratory-built electric unicycle.

Design and Implementation of a Linear Quadratic Regulator Based Maximum Power Point Tracker for Solar Photo-Voltaic System

A maximum power point tracking (MPPT) technique based on linear quadratic regulator (LQR) approach for solar photo-voltaic system has been proposed in this paper. LQR based MPPT controller has been designed with online set-point adjustment approach using current, radiation and temperature sensors. Real time simulations have been carried out on MATLAB TM /dSPACE TM platform for solar photo-voltaic system with buck converter. Performance of the proposed technique hase been compared with perturb and observe, incremental conductance, fuzzy logic, neural network and ANFIS based methods of maximum power point tracking. Experimental results showed the superiority of the proposed method for tracking maximum power point under rapidly varying solar radiations.

A Comparative Analysis of Repetitive and Resonant Controllers to a Servo-Vision Ball and Plate System

This paper presents a comparative analysis of two alternative control strategies based on the Internal Model Principle: the Repetitive Controller and the Multiple Resonant Controller. The proposed implementations are evaluated experimentally in a servo-vision ball and plate balancing system. The plant is composed by an orientable platform with a free rolling sphere on top, where the controlled variable is the ball position. The methodology considered to synthesize the associated state gains parameters is the Linear Quadratic Regulator approach. The experimental results compare characteristics such as steady-state error, transient response and input signal for each implemented control strategy.

Guidance and Control of Position and Attitude for Rendezvous and Dock/Berthing with a Noncooperative/Target Spacecraft

Noncooperative target spacecrafts are those assets in orbit that cannot convey any information about their states (position, attitude, and velocities) or facilitate rendezvous and docking/berthing (RVD/B) process. Designing a guidance, navigation, and control (GNC) module for the chaser in a RVD/B mission with noncooperative target should be inevitably solved for on-orbit servicing technologies. The proximity operations and the guidance for achieving rendezvous problems are addressed in this paper. The out-of-plane maneuvers of proximity operations are explored with distinct subphases, including a chaser far approach in the target’s orbit to the first hold point and a closer approach to the final berthing location. Accordingly, guidance solutions are chosen for each subphase from the standard Hill based Closhessy-Willtshire (CW) solution, elliptical fly-around, and Glideslope algorithms. The control is based on a linear quadratic regulator approach (LQR). At the final berthing location, attitude tracker based on a proportional derivative (PD) form is tested to synchronize the chaser and target attitudes. The paper analyzes the performance of both controllers in terms of the tracking ability and the robustness. Finally, it prescribes any restrictions that may be imposed on the guidance during any subphase which can help to improve the controllers tracking ability.

A Self-Tuning LQR Approach Demonstrated on an Inverted Pendulum

Abstract An automatic controller tuning approach is presented that iteratively updates a linear quadratic regulator (LQR) design such that the resulting controller achieves improved closed-loop performance. In each iteration, an updated LQR gain is obtained by adjusting the weighting matrices of the associated quadratic cost. The performance of the resulting controller (measured in terms of another quadratic cost with fixed weights) is evaluated from experimental data obtained by testing the controller in closed-loop operation. The weight adjustment occurs through a stochastic optimization seeking to minimize the experimental cost. Simulation results of a stochastic linear system show that the self-tuning algorithm can recover optimal performance despite having imprecise model knowledge. Experiments on an inverted pendulum demonstrate that the method is effective in improving the system's balancing performance.

Shunt reactive VAR compensator for grid‐connected induction generator in wind energy conversion systems

This work presents the control design for compensating reactive power requirement of induction generator (IG) in wind generation systems using STATic COMpensator (STATCOM). A mathematical model of IG is developed in synchronously rotating d–q–0 axis. The STATCOM is realised using voltage source inverter for which the switching function model is derived and employed here. Instantaneous p–q theory and symmetrical components theory are considered for reference current generation. The current control uses an optimal proportional controller designed using linear quadratic regulator (LQR) approach. Comparative analysis is also made between hysteresis current control and LQR. Simulation and experimental results indicate that the suggested control techniques make the supply power factor close to unity.

Preliminary Attitude Control Studies for the ASTER Mission

This work discusses an attitude control study for the ASTER mission, the first Brazilian mission to the deep space. The study is part of a larger scenario that is the development of optimal trajectories to navigate in the 2001 SN263 asteroid system, together with the generation of orbit and attitude controllers for autonomous operation. The spacecraft attitude is defined from the orientation of the body reference system to the Local Vertical Local Horizontal (LVLH) of a circular orbit around the Alpha asteroid. The rotational equations of motion involve the dynamic equations, where the three angular speeds are generated from a set of three reaction wheels and the gravitational torque. The rotational kinematics is represented in the Euler angles format. The controller is developed via the linear quadratic regulator approach with output feedback. It involves the generation of a stability augmentation (SAS) loop and a tracking outer loop, with a compensator of desired structure. It was chosen the feedback of the p, q and r angular speeds in the SAS, one for each reaction wheel. In the outer loop, it was chosen a proportional integral compensator. The parameters are tuned using a numerical minimization that represents a linear quadratic cost, with weightings in the tracking error and controls. Simulations are performed with the nonlinear model. For small angle manoeuvres, the linear results with reaction wheels or thrusters are reasonable, but, for larger manoeuvres, nonlinear control techniques shall be applied, for example, the sliding mode control.

A new optimal sliding mode controller design using scalar sign function

This paper presents a new optimal sliding mode controller using the scalar sign function method. A smooth, continuous-time scalar sign function is used to replace the discontinuous switching function in the design of a sliding mode controller. The proposed sliding mode controller is designed using an optimal Linear Quadratic Regulator (LQR) approach. The sliding surface of the system is designed using stable eigenvectors and the scalar sign function. Controller simulations are compared with another existing optimal sliding mode controller. To test the effectiveness of the proposed controller, the controller is implemented on an aluminum beam with piezoceramic sensor and actuator for vibration control. This paper includes the control design and stability analysis of the new optimal sliding mode controller, followed by simulation and experimental results. The simulation and experimental results show that the proposed approach is very effective.

Non-probabilistic reliability method and reliability-based optimal LQR design for vibration control of structures with uncertain-but-bounded parameters

Uncertainty is inherent and unavoidable in almost all engineering systems. It is of essential significance to deal with uncertainties by means of reliability approach and to achieve a reasonable balance between reliability against uncertainties and system performance in the control design of uncertain systems. Nevertheless, reliability methods which can be used directly for analysis and synthesis of active control of structures in the presence of uncertainties remain to be developed, especially in non-probabilistic uncertainty situations. In the present paper, the issue of vibration control of uncertain structures using linear quadratic regulator (LQR) approach is studied from the viewpoint of reliability. An efficient non-probabilistic robust reliability method for LQR-based static output feedback robust control of uncertain structures is presented by treating bounded uncertain parameters as interval variables. The optimal vibration controller design for uncertain structures is carried out by solving a robust reliability-based optimization problem with the objective to minimize the quadratic performance index. The controller obtained may possess optimum performance under the condition that the controlled structure is robustly reliable with respect to admissible uncertainties. The proposed method provides an essential basis for achieving a balance between robustness and performance in controller design of uncertain structures. The presented formulations are in the framework of linear matrix inequality and can be carried out conveniently. Two numerical examples are provided to illustrate the effectiveness and feasibility of the present method.