Classical LQR Research Articles

A Tube-MPC Approach to Autonomous Multi-Vehicle Racing on High-Speed Ovals

Autonomous vehicle racing has emerged as vibrant, innovative technology development, demonstration platform in recent years. Universities, companies demonstrate their achievements on various vehicles - from 1:10th to full-scale prototypes. One of those platforms is the Dallara AV-21, and the spec-vehicle for the Indy Autonomous Challenge. This paper outlines the robust model predictive control (MPC) concept used within the software stack of the TUM Autonomous Motorsport team. It is based on a simplified friction-limited point mass model and a set of low-level feedback controllers. The remaining model uncertainties are managed via introducing a constraint-tightening approach based on a Tube-MPC approach. In contrast to classical tracking controllers, the optimization problem is formulated to freely optimize the trajectory while staying within certain maximum deviations of the reference. This approach allows to rely on a coarse output of the trajectory planning approach while maintaining smoothness requirements in steering, throttle, and brake actuation. The paper highlights the advantages of the proposed robust reoptimization concept compared to pure tracking formulations. It showcases the performance compared to a classical LQR controller and an MPC, which utilizes a vehicle model with a more sophisticated tire model. The controller achieved a top speed of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{265 kmh}^{-1}$</tex-math></inline-formula> and lateral accelerations up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\text{21 ms}}^{-2}$</tex-math></inline-formula> during a two-vehicle competition involving dynamic overtaking maneuvers on the Las Vegas Motor Speedway, a famous racetrack with turns banked up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$20^{\circ }$</tex-math></inline-formula> .

An efficient arc-search interior-point algorithm for convex quadratic programming with box constraints

Although the classical LQR design method has been very successful in real world engineering designs, in some cases, the classical design method needs modifications because of the saturation in actuators. This modified problem is sometimes called the constrained LQR design. For discrete systems, the constrained LQR design problem is equivalent to a convex quadratic programming problem with box constraints. We will show that the interior-point method is very efficient for this problem because an initial interior point is available, a condition which is not true for general convex quadratic programming problem. We will devise an effective and efficient algorithm for the constrained LQR design problem using the special structure of the box constraints and a recently introduced arc-search technique for the interior-point algorithm. We will prove that the algorithm is polynomial and has the best-known complexity bound for the convex quadratic programming. The proposed algorithm is implemented in MATLAB. An example for the constrained LQR design is provided to show the effectiveness and efficiency of the design method. The proposed algorithm can easily be used for model predictive control.

Fast Real-Time Model Predictive Control for a Ball-on-Plate Process

This work is concerned with an original ball-on-plate laboratory process. First, a simplified process model based on state–space process description is derived. Next, a fast state–space MPC algorithm is discussed. Its main advantage is computational simplicity: the manipulated variables are found on-line using explicit formulas with parameters calculated off-line; no real-time optimization is necessary. Software and hardware implementation details of the considered MPC algorithm using the STM32 microcontroller are presented. Tuning of the fast MPC algorithm is discussed. To show the efficacy of the MPC algorithm, it is compared with the classical PID and LQR controllers.

A Study of Anti-swing Fuzzy LQR Control of a Double Serial Link Rotary Pendulum

In this work, a Fuzzy Linear Quadratic Regulator (FLQR) is developed for the anti-swing control problem of a Double Serial Link Rotary Pendulum (DSLRP) system. The main objective of this paper is to study the anti-swing FLQR controller and compare it with the classical LQR controller. A dynamic mechanical model of the DLRP was simulated numerically in the SimMechanics toolbox/MATLAB. This model can be defined mathematically as the single input and multiple output non-linear dynamic equations. To verify the performance of both anti-swing controllers, different time-domain control specifications are computed. Moreover, to show the controllers performance, simulation and experiments results are given and compared. According to the comparative results, the FLQR anti-swing controller produces better results than classical LQR in term of time-domain control specifications. Moreover, the controllers’ dynamic responses were compared according to robustness analysis in the presence of external force disturbance. The results obtained here indicate that the FLQR anti-swing controller product better performance than others in term of vibration suppression capability.

Development of a Fuzzy-LQR and Fuzzy-LQG stability control for a double link rotary inverted pendulum

In this paper, a Fuzzy based Linear Quadratic Regulator (FLQR) and Linear Quadratic Gaussian (FLQG) controllers are developed for stability control of a Double Link Rotary Inverted Pendulum (DLRIP) system. The aim of this work is to study dynamic performance analysis of both FLQR and FLQG controllers and to compare them with the classical LQR and LQG controllers, respectively. A dynamic mechanical simulation model of the DLRIP was obtained using both the numerically SimMechanics toolbox in MATLAB and the analytically dynamic nonlinear equations. To determine the control performance of the controllers, Settling Time (Ts), Peak Overshoot (PO), Steady-State Error (Ess), and the total Root Mean Squared Errors (RMSEs) of the joint positions are computed. Furthermore, the dynamic responses of the controllers were compared based on robustness analysis under internal and external disturbances. To show the control performance of the controllers, several simulations were conducted. Based on the comparative results, the dynamic responses of both FLQR and FLQG controllers produce much better results than the dynamic responses of the classical LQR and LQG controllers, respectively. Moreover, the robustness results indicate that the FLQR and FLQG controllers under the internal and external disturbances were effective.

Deep Reinforcement Learning with embedded LQR Controllers

Reinforcement learning is a model-free optimal control method that optimizes a control policy through direct interaction with the environment. For reaching tasks that end in regulation, popular discrete-action methods are not well suited due to chattering in the goal state. We compare three different ways to solve this problem through combining reinforcement learning with classical LQR control. In particular, we introduce a method that integrates LQR control into the action set, allowing generalization and avoiding fixing the computed control in the replay memory if it is based on learned dynamics. We also embed LQR control into a continuous-action method. In all cases, we show that adding LQR control can improve performance, although the effect is more profound if it can be used to augment a discrete action set.

A State Feedback Controller Based on GCC Algorithm against Wind‐Induced Motion for High‐Rise Buildings with Parametric Uncertainties

High‐rise buildings with an active mass damper/driver (AMD) system generally use a simplified mathematical model. The result leads to parametric uncertainties including the stiffness and mass variations exist, and the accuracy of its controller is seriously affected. In view of the above uncertainties, based on a Lyapunov stability theory and linear matrix inequality (LMI) approach, a state feedback controller based on the guaranteed cost control (GCC) algorithm is proposed in this paper. A ten‐storey frame with an AMD system is taken as a numerical example, and its control effect and AMD parameters are viewed as the control indexes. The performance of the proposed robust controller is compared with a conventional controller based on the classical LQR algorithm. The analysis results show that the new controller effectively suppresses the dynamic responses of high‐rise buildings with parametric uncertainties and keeps the AMD parameters stable. Finally, a four‐storey steel experimental frame with an AMD system is used to verify the correctness of the conclusions.

Design of an optimal state derivative feedback LQR controller and its application to an offshore steel jacket platform

This paper concerns with the optimal state derivative feedback LQR controller design for vibration control of an offshore steel jacket platform having active tuned mass damper against the wave induced disturbances. Considering that the state derivative signals such as acceleration and velocity are easier to measure rather than the state variables such as displacement, state derivative feedback control strategy is proposed to obtain practically applicable and easily realizable synthesis method. On the basis of convex optimization approach, state derivative feedback LQR controller design is formulated in Linear Matrix Inequalities (LMIs) form to get an optimal feasible solution set. Finally, an offshore steel jacket platform subject to nonlinear self excited wave force is used to illustrate the effectiveness of the proposed approach through simulations. The results show that proposed state derivative LQR controller is very effective in reducing vibration amplitudes of each floor of modeled offshore steel jacket platform and achieves compitable control performance to classical LQR controller design.

Some Philosophical Paradigms in Education of Modeling and Control

The paper discusses some interesting, mainly philosophical paradigms of the modeling and control areas, which are still partly unsolved and/or only partially studied. First the possible introduction of a prejudice free control - similar to the term for the modeling introduced by Rudi Kalman - is investigated. Next the real constraints in real control systems are discussed. It seems that these are the amplitude limit for the actuators in practical systems. Then the application of the HEISENBERG-type uncertainty relationship in control is discussed, combining the robustness of control and quality of modeling. Finally a special irregularity in the classical LQR control methodology is treated investing some unreachable poles in the anomaly. The paper constitutes a review of a few specific problems in the control theory. They are enumerated in keywords and discussed on the basis of the references. The authors aim at least to invite further authors to continue such a discussion.

Guaranteeing Constraints of Disturbed Nonlinear Systems Using Set-Based Optimal Control in Generator Space

We address the problem of finding an optimal solution for a nonlinear system for a set of initial states rather than just for a single initial state. In addition, we consider state and input constraints as well as a set of possible disturbances. While previous optimal control techniques typically ignore the fact that the current state of a system is not exactly known, future safety-critical systems demand that all uncertainties including the initial state are considered; this is required for e.g. automated vehicles, surgical robots, or human-robot interaction. We present a new method that obtains optimal control inputs by finding optimal weights for generators that span the space reachable by the considered system. This solution routine can be used not only for a single initial state but also for a set of initial states - this is not possible using classical optimization techniques. We ensure that all constraints are met by using reachability analysis, which provides formal bounds for all possible system trajectories. We demonstrate the applicability of our approach with an example from automated driving; for this example, the result is obtained within a few seconds and outperforms a classical LQR approach.

HEX2OQTAL: Translational Optimal Control Exploiting Quaternion Error Dynamics

The work presented in this paper is a ground-breaking initiative to design a translational optimal controller named HEX2OQTAL. The controller is capable of producing high precision translational tracking using a linearized model of a trajectory error based on a quaternion attitude representation. This control contribution is general in the sense that it is agnostic of the underlying vehicle; the results are presented for an unmanned aerial vehicle (UAV) experimentally, a spacecraft scenario in simulation, and a precision planetary landing scenario. The paper presents a full derivation from base principles, a proof of stability, and real-time repeatable experimental UAV results (25 tests per controller per scenario). These results show that the developed controller presented here is far more precise than the most common industry standard controllers, such as PID and classical LQR. The obtained results show the potential of this controller not only as an accurate and adaptable controller, but also in high precision trajectory tracking control.

Observer Based Optimal Vibration Control of a Full Aircraft System Having Active Landing Gears and Biodynamic Pilot Model

This paper deals with the design of an observed based optimal state feedback controller having pole location constraints for an active vibration mitigation problem of an aircraft system. An eleven-degree-of-freedom detailed full aircraft mathematical model having active landing gears and a seated pilot body is developed to control and analyze aircraft vibrations caused by runway excitation, when the aircraft is taxiing. Ground induced vibration can contribute to the reduction of pilot’s capability to control the aircraft and cause the safety problem before take-off and after landing. Since the state variables of the pilot body are not available for measurement in practice, an observed based optimal controller is designed via Linear Matrix Inequalities (LMIs) approach. In addition, classical LQR controller is designed to investigate effectiveness of the proposed controller. The system is then simulated against the bump and random runway excitation. The simulation results demonstrate that the proposed controller provides significant improvements in reducing vibration amplitudes of aircraft fuselage and pilot’s head and maintains the safety requirements in terms of suspension stroke and tire deflection.

Self-triggered linear quadratic control

Self-triggered control is a recently proposed paradigm that abandons the more traditional periodic time-triggered execution of control tasks with the objective of reducing the utilization of communication resources, while still guaranteeing desirable closed-loop behavior. In this paper, we introduce a self-triggered strategy based on performance levels described by a quadratic discounted cost. The classical LQR problem can be recovered as an important special case of the proposed self-triggered strategy. The self-triggered strategy proposed in this paper possesses three important features. Firstly, the control laws and triggering mechanisms are synthesized so that a priori chosen performance levels are guaranteed by design. Secondly, they realize significant reductions in the usage of communication resources. Thirdly, we address the co-design problem of jointly designing the feedback law and the triggering condition. By means of a numerical example, we show the effectiveness of the presented strategy. In particular, for the self-triggered LQR strategy, we show quantitatively that the proposed scheme can outperform conventional periodic time-triggered solutions.

Optimal Control of Slow-Active Vehicle Suspension — Results of Experimental Data

Laboratory investigations of an active vehicle suspension of an in-series structure of the slow-active type are presented in this paper. The multidimensional model is reduced to a case representing quarter car suspension. Control laws for active vibration reduction systems are usually determined based upon linear models of objects. Active suspensions are characterised by nonlinearity, connected most often with actuating systems and their energetic restrictions. This causes divergences between theoretical quality factors and those determined experimentally. The second essential problem is finding a compromise between opposed quality factors (for example, minimum power requirement and high efficiency of vibration reduction). Thanks to the use of the proper control law in the active vibration reduction system of vehicle suspension, the goal of ensuring a high level of ride comfort, good vehicle handling and incessant contact of the wheels with the road surface with a minimum power requirement may be attained. The authors, in seeking a compromise, determined classical LQR controllers for the proposed quality indicators realising the aims mentioned above. These controllers are determined for a linearised suspension model obtained from identification. Experimental characteristics are determined for all of the suspension control systems.

Semiactive Wind Response Control of 76-Story Benchmark Building with Smart Piezoelectric Friction Dampers

A smart piezoelectric friction damper (SPFD) is presented based on improved Pall friction dampers and its damping force model is analyzed. The wind-induced vibration control of Benchmark model using semi-active control strategies based on the classical LQR is studied. The results of simulation analysis show that the semi-active control effects of the standard wind-control model with SPFD is evident. Compared to uncontrolled structures, the wind-induced vibration responses of the controlled structures are effectively reduced. In addition, parameter optimization of the semi-active control system based on the limit Hrovat optimal control algorithm is carried out. The analysis shows that the robustness of the semi-active control system to the stiffness uncertainty of Benchmark Model is very good, but the robustness to the damping uncertainty is not so good.

Fuzzy-sliding mode control of a full car semi-active suspension systems with MR dampers

A fuzzy-sliding mode controller is presented to control the dynamics of semi-active suspension systems of vehicles using magneto-rheological (MR) fluid dampers. A full car model is used to design and evaluate the performance of the proposed semi-active controlled suspension system. Four mixed mode MR dampers are designed, manufactured, and integrated with four independent sliding mode controllers. The siding mode controller is designed to decrease the energy consumption and maintain robustness. In order to overcome the chattering of the sliding mode controllers, a fuzzy logic control strategy is merged into the sliding mode controller. The proposed fuzzy-sliding mode controller is designed and fabricated. The performance of the semi-active suspensions is evaluated in both the time and frequency domains. The obtained results demonstrate that the proposed fuzzy-sliding mode controller can effectively suppress the vibration of vehicles and improve their ride comfort and handling stability. Furthermore, it is shown that the "chattering" of the sliding mode controller is smoothed when it is integrated with a fuzzy logic control strategy. Although the cost function of the fuzzy-sliding mode control is a slightly higher than that of a classical LQR controller, the control effectiveness and robustness are enhanced considerably.

Virtual Reality Simulation of Active Car Suspension System

This paper presents the design of a Nonlinear Energy Sink (NES) controller and its application to active sus-pension systems in the Virtual Reality Environment. In this en-vironment, the design engineers are immersed in an audio-visually coupled tele-operated environment whereby direct in-teraction with and control of the design process is achieved in real time. In this manner, the behavior of synthetic models of the full car can be monitored by literally walking around the car and adjusting the design parameters of the suspension as needed to ensure optimal performance while satisfying design and operational requirements. The control actuators which provide forces equivalent to nonlinear stiffness and damping elements are attached to the vehicle in order to actively isolate it from road excitation. The effect of the parameters of the NES controller on the vehicle performance is studied both in the frequency and time domain. The effectiveness of the NES controller is validated by numeri-cal simulation. The robustness of the nonlinear energy pumping process is studied by varying the magnitude of road excitation. The simulation results in the Virtual Reality Environment show that under certain conditions, the nonlinear energy pumping can be induced and significant vibration isolation can be achieved. The performance of vehicle including the ride com-fort and road holding capability can be improved significantly. When the magnitude of road excitation is increased, the capaci-ty of the NES to absorb energy from the main system is also enhanced. This is very important to achieve vibration isolation objectives. The virtual reality simulation results also show that the nonlinear NES controller performs better than the classical LQR controller particularly as the road condition becomes worst.

Optimal control of a pretwisted shearable smart composite rotating beam

The optimal vibration control of a rotating, composite, pretwisted, single-celled box beam, exhibiting transverse shear flexibility and restrained warping, is analyzed. A higher-order shear deformation theory (HSDT) enabling satisfaction of traction-free boundary conditions is employed. An orthotropic host with Circumferentially Uniform Stiffness ply angle configuration and transversely isotropic sensors-actuator pairs that are surface embedded along the span are considered. The total output from sensors is fed to a controller and then uniformly applied to actuators. The extended Galerkin method, along with either instantaneous or classical LQR methods, is used. Instantaneous LQR provides greater response attenuation in case of sustained external forcing. Results are obtained for a linear spanwise variation of pretwist. Compared to the unshearable and first-order shearable (FSDT) models, the HSDT appears most sensitive to pretwist and – when a saturation constraint is considered – it predicts the lowest settling time. The HSDT predicts significant attenuation in response and power required as compared to the FSDT, the differences being especially pronounced for constrained input control. Parametric studies involving the ply angle, pretwist, and patch length are performed. An optimum pretwist that yields lowest response, power, and settling time is obtained. Its value differs when the saturation constraint is used. Tailoring provides greater attenuation at the expense of an increase in settling time. Using constrained input control, an order-of-magnitude reduction in power requirement is possible via tailoring. The results underscore the importance of shear strain variation across the beam wall, and also of synthesizing active control with tailoring, for achieving efficient control.

Optimal two-degree-of-freedom fuzzy control for locomotion control of a hydraulically actuated hexapod robot

Locomotion control of legged robots is a very challenging task because very accurate foot trajectory tracking control is necessary for stable walking. An electro-hydraulically actuated walking robot has sufficient power to walk on rough terrain and carry a heavier payload. However, electro-hydraulic servo systems suffer from various shortcomings such as a high degree of nonlinearity, uncertainty due to changing hydraulic properties, delay due to oil flow and dead-zone of the proportional electromagnetic control valves. These shortcomings lead to inaccurate analytical system model, therefore, application of classical control techniques result into large tracking error. Fuzzy logic is capable of modeling mathematically complex or ill-defined systems. Therefore, fuzzy logic is becoming popular for synthesis of control systems for complex and nonlinear plants. In this investigation, a two-degree-of-freedom fuzzy controller, consisting of a one-step-ahead fuzzy prefilter in the feed-forward loop and a PI-like fuzzy controller in the feedback loop, has been proposed for foot trajectory tracking control of a hydraulically actuated hexapod robot. The fuzzy prefilter has been designed by a genetic algorithm (GA) based optimization. The prefilter overcomes the flattery delay caused by the hydraulic dead-zone of the electromagnetic proportional control valve and thus helps to achieve better tracking. The feedback fuzzy controller ensures the stability of the overall system in the face of model uncertainty associated with hydraulically actuated robotic mechanisms. Experimental results exhibit that the proposed controller manifests better foot trajectory tracking performance compared to single-degree-of-freedom (SDF) fuzzy controller or optimal classical controller like state feedback LQR controller.

Nonlinear control for magnetic levitation of automotive engine vales

Position regulation of a magnetic levitation device is achieved through a control Lyapunov function (CLF) feedback design. It is shown experimentally that by selecting the CLF based on the solution to an algebraic Riccati equation it is possible to tune the performance of the controller using intuition from classical LQR control. The CLF is used with Sontag's universal stabilizing feedback to provide enhanced transient performance farther away from the origin than was achieved with the LQR controller.