State Feedback Linear Quadratic Regulator Research Articles

Enhancing Mechanical Safety in Suspension Systems: Harnessing Control Lyapunov and Barrier Functions for Nonlinear Quarter Car Model via Quadratic Programs

Limiting the suspension stroke in vehicles holds critical and conceivable benefits. It is crucial for the safety, stability, ride comfort, and overall performance of the vehicle. Furthermore, it improves the reliability of suspension components and maintains consistent handling during regular and rough driving conditions. Hence, the design of a safety-critical controller to limit the suspension stroke for active suspension systems is of high importance. In this study, we employed a quarter-car model that incorporates a suspension spring with cubic nonlinearity. The proposed safety-critical controller is the control Lyapunov function–control barrier function–quadratic programming (CLF-CBF-QP). Initially, we designed the reference controller as a linear quadratic regulator (LQR) controller based on the linearized quarter-car model. The reference state-feedback LQR controller simplified the design of the control Lyapunov function. Consequently, from the nonlinear model, we construct a simple control Lyapunov function that relies only on the sprung mass velocity to have a relative degree of one. The CLF intends to improve the performance by considering the nonlinearity and via online optimization. We then formulate the control barrier function to restrict the suspension stroke from breaching its limits. To assess the effectiveness of the proposed controller, we present two challenging road inputs for the nonlinear quarter-car model when employing CLF-CBF-QP and LQR controllers. The CLF-CBF-QP findings surpassed the LQR controller in terms of safety and performance. This study highlights the immense potential of CLF-CBF-QP for suspension systems, improving the time-domain performance, limiting the suspension stroke, and guaranteeing safety.

Robust MPC-RG for an autonomous racing vehicle considering obstacles and the battery state of charge

The design of a controller able to deal with uncertainties and physical constraints plays an essential role in fast and complex systems. Then, a reference governor approach based on model predictive control (MPC-RG) for an autonomous racing vehicle is proposed. The MPC-RG guarantees constraint satisfaction and recursive feasibility online while including obstacle avoidance capability and energy-aware management by solving a multi-objective optimization problem. In particular, a trade-off between maximizing the longitudinal velocity and the state of charge of the vehicle’s battery, as well as minimizing the variation of control actions is adopted. Moreover, the proposed MPC-RG is combined with a state-feedback linear quadratic regulator (LQR) and a Kalman filter (KF) to compensate for modeling errors and exogenous disturbances, as well as to estimate the unmeasured lateral velocity. In fact, for control and estimation purposes, a data-driven Takagi–Sugeno (TS) model trained by an adaptive neuro-fuzzy inference network is used. The performance of the developed approach is assessed in simulations using a well-known case study based on a 1/10 scale RC electric car.

Brake Current Control System Modeling Using Linear Quadratic Regulator (LQR) and Proportional integral derivative (PID)

This paper provides a comparative analysis between PID control as a classical control technique and modern control technique in the dinamometer Eddy current brakes system. Eddy current brakes is a modern braking system that requires a control system to support the braking performance. PID control is often used to be implemented but in some conditions it is less optimal. Therefore, it is necessary to develop a modern and optimal control, such as a full state feedback Linear Quadratic Regulator (LQR). The comparison of the braking time responses were simulated using Matlab/Simulink. The simulation results show that the response of LQR control is better than the PID, with Ts = 2.12 seconds, Tr = 1.18 seconds, and without overshoot. On the other side, PID control, although having Ts = 0.27 seconds and Tr = 0.18 seconds, there is still an overshoot about 0.7%.

H∞/LQR Optimal Control for a Supersonic Air-Breathing Missile of Asymmetric Configuration

Abstract Robust control is challenging to achieve for air-breathing missiles operating in a high Mach number regime, such as at high supersonic speeds (M > 3). The challenge arises because of strong couplings, significant non-linearities and large uncertainties in the aerodynamics and propulsion system. The feasibility of achieving robust control in such applications is strongly linked to the development of an appropriate control design structure. The purpose of this paper is to illustrate that in order to stabilise a highly unstable airframe and achieve the required performance, a hybrid of two control schemes may be used to achieve best results. A state feedback linear quadratic regulator is used to stabilise the plant and a forward path H∞ optimal controller is used to achieve the required performance and robustness. We also highlight the complementary attributes of the two control schemes that together can generate a more robust controller; LQR is used since it can achieve good gain and phase margins, whereas, the H∞ control method is better equipped to deal with uncertainties.

Active structural control of a floating wind turbine with a stroke-limited hybrid mass damper

Floating wind turbines are subjected to more severe structural loads than fixed-bottom wind turbines due to additional degrees of freedom (DOFs) of their floating foundations. It's a promising way of using active structural control method to improve the structural responses of floating wind turbines. This paper investigates an active vibration control strategy for a barge-type floating wind turbine by setting a stroke-limited hybrid mass damper (HMD) in the turbine's nacelle. Firstly, a contact nonlinear modeling method for the floating wind turbine with clearance between the HMD and the stroke limiters is presented based on Euler-Lagrange's equations and an active control model of the whole system is established. The structural parameters are validated for the active control model and an equivalent load coefficient method is presented for identifying the wind and wave disturbances. Then, a state-feedback linear quadratic regulator (LQR) controller is designed to reduce vibration and loads of the wind turbine, and two optimization methods are combined to optimize the weighting coefficients when considering the stroke of the HMD and the active control power consumption as constraints. Finally, the designed controllers are implemented in high fidelity simulations under five typical wind and wave conditions. The results show that active HMD control strategy is shown to be achievable and the designed controllers could further reduce more vibration and loads of the wind turbine under the constraints of stroke limitation and power consumption. “V”-shaped distribution of the TMD suppression effect is inconsistent with the Weibull distribution in practical offshore floating wind farms, and the active HMD control could overcome this shortcoming of the passive TMD.

Construction Tower Crane SimMechanics-Visualized Modelling, Tower Vibration Impact on Payload Swing Analysis and LQR Swing Control

Fast and accurate positioning and swing minimization of payloads in tower crane operation are challenging as well as conflicting tasks. Minimizing the load swing is primarily limited by the existence of wind disturbance effect triggering higher tower vibration, especially in the high speed course and load is in the end of tower region. In this paper, the authors propose a new modeling scheme for 3D tower crane, analyze wind disturbance affected-tower vibration impact on the load swing and develop LQR swing control. Firstly, SimMechanics-visualized crane model has been developed using actual tower crane Morrow (Liebherr 71EC) datasheet. Secondly, wind disturbance model was designed based on Gawronski approach and applies different wind patterns on the model for vibration impact analysis. Thirdly, improved version of linear least square system identification algorithm using 7 past inputs-outputs data has been developed to create better fit model. And finally, propose a new control scheme for 3D tower crane that consists of a reference point tracking with full state feedback Linear Quadratic Regulator (LQR). The simulations results show the robust controller performs well in minimizing the load swing due to wind-affected tower vibration and high speed region.

Optimal Control of Electromagnetic Suspension EMS System

This paper presents the controller design of magnetic levitation application. The highly nonlinear electromag- netic suspension EMS system is hard and limited system control subjected to prescribed stability of system. Due to the nonlinear dynamics of system, the linearization of the nonlinear EMS plant is described by linear model. An attraction force about the prescribed nominal operating point of current and air gap positioning is chosen for linearization by a nominal operating point. Optimal control is applied for controlling nonlinear dynamics of EMS plant. Linear quadratic regulator (LQR) controller applies for this control objective. The system stability is proofed by using Lyapunov's method. From the results the reference of air gap position can be tracked with the desired nominal operating control as shown in simulation and practical manner. Electromagnetic suspension EMS system is significantly applied in advanced technology such as ground transporta- tion, magnetic bearing and other applications. Control of electromagnetic suspension EMS system is a key technology but has some difficulties due to its nonlinearity, instability, and uncertainty. To overcome the problems mentiond. This research focuses on designing the control system for increas- ing the performance of EMS system. Normally many re- searchers applied the nonlinear control for EMS system. From references (1) proposed nonlinear state and output- feedback H ! controllers to suppress guide way induced dis- turbances. From experiment result, air-gap position can be controlled at 4 mm at nominal operating point. (2) also pro- posed the designed linear quadratic state feedback regulator. It can maintain the closed-loop stability in the presence of some certain actuator failures. The control of such a system is applied to fault control application of electromagnets sus- pension model of maglev train. The simulation study of the design is done by numerical study of magnetic levitation model in linear model. (3) presented the results on the robust stabilization of a class of feedback linearized nonlinear sin- gle-input/single-output (SISO) systems with parametric un- certainty. They implemented of nonlinear feedback lineariz- ing control for an electromagnetic suspension EMS system. Comparison on the technique of feedback linearization and classical state feedback control by using linearization with small perturbation is demonstrated by practical experiment. They concluded that feedback linearization control yielded the stable control of tracking at air gap reference, while the classical state feedback control using linearization at small perturbation could not keep system stable. The model of dy- namics and control of electromagnetic suspension is de- scribed by (4). And (5) also proposed the applied magnetic levitation for educational purpose. They described the opera- tion of a pulse width modulation converter in a magnetic suspension system. The pulse width-modulated (PWM) con- verter illustrates modern principles of power electronics, such as PWM control, current-mode control, averaged and linearized models of switched-mode converters, and power supply design. The experimental system shown the levitation control with diameter 6 cm and weight 0.8 kg of a metallic sphere. (6) applied the laser displacement measurement for the feedback control of air gap of magnetic levitation and suspension system. Then system is increased the accuracy and smoothness of control output signal due to the high reso- lution of feedback measurement device.

Improving Stability Margin in Electric Power Systems by Linear Quadratic Regulator and Disturbance Model

This article deals with the design of a voltage controller using state feedback with integral action for an electric power system. The control system is tailored to achieve zero error of the voltage magnitude. First the transient response and the attraction region of the non-linear model is improved by using a state-feedback linear quadratic regulator controller. On the basis of this design, the integral action is designed as a second loop according to the internal model controller. Both feedback control systems are compared in terms of the robustness of the closed-loop dynamic response and the existing coupling between active and reactive powers when the load reactive power is changed. Simulation results show that the integral controller achieves zero voltage error even for the non-linear system.

Stability for switched dynamic hybrid systems

This paper investigates quadratic stabilizability for switched dynamic hybrid systems with polytopic uncertainties via a common Lyapunov matrix or via closed loop linear quadratic state feedback regulators. The switched linear systems for stable closed loop matrices can guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.

Conditions for stabilizability of linear switched systems

This paper investigates some conditions that can provide stabilizability for linear switched systems with polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive a Linear Matrix Inequality (LMI) to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.

Digital Modeling and PID Controller Design for MIMO Analog Systems with Multiple Delays in States, Inputs and Outputs

This paper presents a discrete-time state-space methodology for the digital modeling and design of an optimal digital proportional-integral-derivative (PID) plus state-feedback controller for multiple-input, multiple-output (MIMO) continuous-time systems with multiple time delays in states, inputs and outputs. To implement the digital design, first the Chebyshev quadrature formula together with a linear interpolation method is employed to obtain an extended discrete-time state-space model from the continuous-time multiple time-delay system. Then, a partially predetermined digital PID controller and the extended discrete-time state-space model are formulated as an augmented discrete-time state-space system utilizing state-feedforward and state-feedback linear-quadratic regulator (LQR) design. As a result, the parameters of the optimal PID controller and its associated state-feedback controller can be determined by tuning the weighting matrices in the LQR performance criteria. Further, an optimal discrete-time observer is jointly constructed for the multivariable system with multiple delays in states, inputs and outputs. The proposed design methodology can be applied to general MIMO continuous-time multiple time-delay systems for performance improvement and disturbance rejection. An illustrative example is given to demonstrate the effectiveness of the developed method.

Impact Response of Adaptive Piezoelectric Laminated Plates

Semi-analytical models for the impact response of composite plates having distributed active and sensory piezoelectric layers are presented. An exact Ritz model is utilized based on a mixed-e eld piezoelectric laminate theory thatencompassesbothdisplacementandelectricdisplacemente eldsthroughthelaminate.Thegoverningequations of motion of a simply supported plate and a low-velocity impactor are formulated including impactor dynamics and contact law. State feedback linear quadratic regulator and output feedback controllers are implemented to actively reduce impact force. Numerical results quantify the electromechanical response of a composite plate with piezoelectric sensors impacted by three different mass impactors. The impact response of an actively controlled plate is also shown, and the possibility to improve critical impact parameters is demonstrated.

Two-time scale controller design for a high speed planar parallel manipulator with structural flexibility

Planar parallel manipulators, with potential applications in high speed, high acceleration tasks such as electronic component placement, would be subject to mechanical vibration due to high inertial forces acting on the linkages and other components. To achieve high throughput capability, such motion induced vibration would have to be damped quickly, to reduce settling time of the platform position and orientation. This paper develops a two-time scale dynamic model of a three-degree-of-freedom planar parallel manipulator with structurally flexible linkages. Based on the two-time scale model, a composite controller, consisting of a computed torque controller for the slow time-scale or rigid body subsystem dynamics, and a linear-quadratic state-feedback regulator for the fast time-scale flexible dynamic subsystem, is designed. Simulation results show that the composite control scheme permits the parallel manipulator platform to follow a given desired trajectory, while damping structural vibration arising due to excitation from inertial forces.

Two-time scale controller design for a high speed planar parallel manipulator with structural flexibility

SummaryPlanar parallel manipulators, with potential applications in high speed, high acceleration tasks such as electronic component placement, would be subject to mechanical vibration due to high inertial forces acting on the linkages and other components. To achieve high throughput capability, such motion induced vibration would have to be damped quickly, to reduce settling time of the platform position and orientation. This paper develops a two-time scale dynamic model of a three-degree-of-freedom planar parallel manipulator with structurally flexible linkages. Based on the two-time scale model, a composite controller, consisting of a computed torque controller for the slow time-scale or rigid body subsystem dynamics, and a linear-quadratic state-feedback regulator for the fast time-scale flexible dynamic subsystem, is designed. Simulation results show that the composite control scheme permits the parallel manipulator platform to follow a given desired trajectory, while damping structural vibration arising due to excitation from inertial forces.

Analysis of robust pole clustering in a sector region for uncertain LQR control systems

This paper presents a general analysis of robust pole clustering in a sector region for uncertain H 2 optimal linear quadratic regulator (LQR) control systems. Control system poles located in a sector region of the left half-plane provide a good performance, such as the transient behavior, overshoot, damping ratio and so on, for control systems. The general Lyapunov theory and Rayleigh principle along the norm theory are applied to analyze robust pole clustering. The H 2 optimal state feedback LQR control systems are considered. The concerned uncertainties are both unstructured and structured uncertainties, including interval matrices. The preservation of H 2 optimality of the LQR in face of uncertainties is discussed. The results are useful for analysis and design of robust control combining robust stability and robust performance.

Stabilization of perturbed systems via linear optimal regulator

Linear quadratic state feedback regulators make the resulting closed-loop systems stable enough, i.e. they realize robust stabilization. Many attempts at robust stabilization using linear quadratic regulators have been reported. One of the major trends of formulating uncertainty in systems is to express perturbed parameters as the sum of two terms, i.e. nominal values and the deviation from them. In this paper, it is assumed that the upper and lower bounds for each uncertain parameter can somehow be estimated. This enables us to dispense with nominal values. The main aim is to contrive a robust feedback stabilization law for systems with parameters falling into certain ranges via a linear quadratic regulator based only upon information on their bounds. The systems under consideration are therefore those having interval system matrices (in which each element has the above-mentioned two-sided bounds). A certain feedback law is a stabilizing law for a system with an interval system matrix if and only if the ...

Robust stability of linear quadratic state feedback regulator under system uncertainty

This paper is concerned with the robustness of LQ state feedback regulators in the presence of system uncertainty. Assuming that the perturbations satisfy the matching condition, several robust stability results are explicitly derived for linear, non-linear, time-varying, and dynamical perturbations in terms of bounds of perturbations and the weighting matrices in the performance index. Some examples are included to show the applicability and the limitation of the present results.

On improving the robustness of LQ regulators

Soroka and Shaked have shown that a linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system can lead to fast unstable closed-loop modes. It is shown there that the stability properties can be improved if possible nonminimum phase behavior is allowed for in modeling the system prior to the controller design.

On the robustness of LQ regulators

It is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed-loop modes.

Output feedback pole placement in the design of suboptimal linear quadratic regulators†

A design oriented methodology is presented for the construction of low-order, suboptimal, output feedback compensators. The design is suboptimal in the sense that it retains an l-dimensional eigenspace from a reference optimal state-feedback linear quadratic regulator problem. If r is the number of outputs, then for fixed l > r, it is known that the design requires a dynamic compensator of dimension l — r. The parameters of the compensator are determined by the solution of an associated output feedback pole placement problem. Using iterative, dyadic pole placement, a procedure is given which determines the solution of this problem, without a priori assumptions on the dimension of the compensator. The design is also extended to the class of stabilizable systems, and the resulting compensator is shown to possess a separation property.