scholarly journals Autonomous Collision Avoidance Using MPC with LQR-Based Weight Transformation

Sensors ◽  
2021 ◽  
Vol 21 (13) ◽  
pp. 4296
Author(s):  
Shayan Taherian ◽  
Kaushik Halder ◽  
Shilp Dixit ◽  
Saber Fallah
Keyword(s):  
Collision Avoidance ◽  
Discrete Time ◽  
Optimal Solution ◽  
Linear Quadratic Regulator ◽  
Weighting Factor ◽  
Algebraic Riccati Equation ◽  
Control Technique ◽  
Linear Quadratic ◽  
Weighting Matrix ◽  
Quadratic Tracking

Model predictive control (MPC) is a multi-objective control technique that can handle system constraints. However, the performance of an MPC controller highly relies on a proper prioritization weight for each objective, which highlights the need for a precise weight tuning technique. In this paper, we propose an analytical tuning technique by matching the MPC controller performance with the performance of a linear quadratic regulator (LQR) controller. The proposed methodology derives the transformation of a LQR weighting matrix with a fixed weighting factor using a discrete algebraic Riccati equation (DARE) and designs an MPC controller using the idea of a discrete time linear quadratic tracking problem (LQT) in the presence of constraints. The proposed methodology ensures optimal performance between unconstrained MPC and LQR controllers and provides a sub-optimal solution while the constraints are active during transient operations. The resulting MPC behaves as the discrete time LQR by selecting an appropriate weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. In this paper, the effectiveness of the proposed technique is investigated in the application of a novel vehicle collision avoidance system that is designed in the form of linear inequality constraints within MPC. The simulation results confirm the potency of the proposed MPC control technique in performing a safe, feasible and collision-free path while respecting the inputs, states and collision avoidance constraints.

scholarly journals Transformation of LQR Weights for Discretization Invariant Performance of PI/PID Dominant Pole Placement Controllers

Robotica ◽  
2019 ◽  
Vol 38 (2) ◽  
pp. 271-298 ◽  
Author(s):  
Kaushik Halder ◽  
Saptarshi Das ◽  
Amitava Gupta
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Pid Controller ◽  
Closed Loop ◽  
Weighting Factor ◽  
Pole Placement ◽  
Algebraic Riccati Equation ◽  
Linear Quadratic ◽  
Similar Time ◽  
Dominant Pole

SummaryLinear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.

scholarly journals Robot Inverted Pendulum Beroda Dua (IPBD) dengan Kendali Linear Quadratic Regulator (LQR)

2019 ◽  
Vol 7 (2) ◽  
pp. 224
Author(s):  
FAHMIZAL FAHMIZAL ◽  
MUHAMMAD ARROFIQ ◽  
RONALD ADRIAN ◽  
AFRIZAL MAYUB
Keyword(s):  
Inverted Pendulum ◽  
Linear Quadratic Regulator ◽  
Control Technique ◽  
System Response ◽  
Linear Quadratic ◽  
Weighting Matrix ◽  
Lqr Control ◽  
Wheeled Inverted Pendulum ◽  
Arduino Uno ◽  
Embedded Programming

ABSTRAKMakalah ini memaparkan proses pemodelan robot inverted pendulum beroda dua (IPBD) menggunakan dinamika Lagrange. Setelah sistem model robot IPBD diperoleh, teknik kendali optimal dalam hal ini menggunakan linear quadratic regulator (LQR) digunakan untuk melihat step respon sistem dan tanggapan respon sistem terhadap gangguan. Sebelum kendali LQR diimplementasikan, simulasi menggunakan Simulink Matlab dilakukan untuk mendapat parameter gain K pada kendali LQR. Selanjutnya, dengan mengubah-ubah matriks pembobot Q akan diperoleh variasi gain K. Pada penelitian ini dilakukan variasi matriks pembobotan Q sebanyak lima jenis. Sedangkan matriks elemen R dituning dengan nilai satu. Dari hasil pengujian diperoleh bahwa dengan membesarkan pembobotan matriks Q, dihasilkan respon menuju keadaan steady lebih cepat dan overshoot berkurang. Parameter gain K dari hasil simulasi selanjutnya akan diimplementasikan secara embedded programming ke dalam Arduino Uno pada sistem robot IPBD.Kata kunci: Inverted pendulum beroda, Pemodelan, LQR ABSTRACTThis paper describes the process of modeling two-wheeled pendulum inverted robots (IPBD) using the Lagrange dynamics. After the IPBD robot model system was obtained, the optimal control technique in this case using a linear quadratic regulator (LQR) was used to see the system response step and the response of the system response to interference. Before the LQR control is implemented, simulation using Matlab Simulink is conducted to get the gain K parameter on the LQR control. Furthermore, by varying the weighting matrix Q, the gain variation K will be obtained. There are five types of Q weighting matrix in this research and the R element matric is tuned with a value of 1. From the test, obtained results show that by raising the weighting matrix Q is produced a faster response to the steady state and overshoot is reduced. At the final stage, the gain K parameter from the simulation results will be implemented by embedded programming into Arduino Uno on the IPBD robot system.Keywords: Wheeled inverted pendulum, Modelling, LQR

Discrete Time Linear Quadratic Tracking Controller for Omni-Directional Mobile Robots

2011 ◽  
Author(s):  
Ehsan Hashemi ◽  
Maani Ghaffari Jadidi
Keyword(s):  
Control System ◽  
Path Planning ◽  
Mobile Robots ◽  
Controller Design ◽  
Optimal Solution ◽  
Linear Quadratic Regulator ◽  
Reference Trajectory ◽  
Linear Quadratic ◽  
Tracking Controller ◽  
Quadratic Tracking

The purpose of this investigation is to suggest and examine a trajectory follower control system for linear discrete dynamic model of omni-directional mobile robots to reach a controller with optimal inputs for drivers. Introducing optimal controllers for multi input-multi output control systems in acceleration and deceleration maneuvers to track a specified path is one of essential subjects for motion study of omni-directional mobile robots. Regulated drivers’ rotational velocities and torques greatly affect the ability of these robots to perform trajectory planner tasks. Moreover, environmental influencing factors shall also be considered in such robot models for accurate path planning. Presented tracking control system in this article provides an optimal solution to minimize differences between reference trajectory and system output in the lately developed simulated model. Trajectory following system together with implemented kinematic and dynamic modeling for an optimal controller to satisfy the path planning prerequisites is mainly discussed in this paper in several sections. Main topics presented and discussed in this article are considerable improvements in simulation of the newly optimized controller by Linear Quadratic Regulator and Tracking. Utilizing the new approach on tracking controller design results in the more precise and appropriate tracking behavior of omni-directional mobile robots as the simulation and experimental results confirm this issue.

Experimental Study of NMP Sample and Hold Input Using an Inverted Pendulum

2018 ◽  
Author(s):  
Yingxu Wang ◽  
Guoming G. Zhu ◽  
Ranjan Mukherjee
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Inverted Pendulum ◽  
Linear Quadratic Regulator ◽  
Sampling Period ◽  
Minimum Phase ◽  
Discrete Time System ◽  
Linear Quadratic ◽  
Sample And Hold ◽  
Control Configuration

Early research showed that a zero-order hold is able to convert a continuous-time non-minimum-phase (NMP) system to a discrete-time minimum-phase (MP) system with a sufficiently large sampling period. However the resulting sample period is often too large to adequately cover the original NMP system dynamics and hence not suitable for control application to take advantage of a discrete-time MP system. This problem was solved using different sample and hold inputs (SHI) to reduce the sampling period significantly for MP discrete-time system. Three SHIs were studied analytically and they are square pulse, forward triangle and backward triangle SHIs. To validate the finding experimentally, a dual-loop linear quadratic regulator (LQR) control configuration is designed for the Quanser single inverted pendulum (SIP) system, where the SIP system is stabilized using the Quanser continuous-time LQR (the first loop) and an additional discrete-time LQR (the second loop) with the proposed SHIs to reduce the cart oscillation. The experimental results show more than 75% reduction of the steady-state cart displacement variance over the single-loop Quanser controller and hence demonstrated the effectiveness of the proposed SHI.

Calculation of Optimized Movement Trajectories for the Human Forearm

2013 ◽  
Vol 401-403 ◽  
pp. 1347-1352
Author(s):  
Li Li Yang
Keyword(s):  
Discrete Time ◽  
Linear Quadratic Regulator ◽  
Minimum Variance ◽  
Movement Trajectory ◽  
Linear Quadratic ◽  
Variance Model ◽  
Movement Trajectories ◽  
Human Forearm ◽  
Optimal Movement ◽  
Time Linear

Using the minimum variance model, optimal human forearm trajectories formation was investigated using a discrete time linear quadratic regulator. First, the continuous dynamics of the human forearm were established on the basis of the relation between muscle torque and neural control signal, and then we transferred the continuous system dynamics to discrete time notation. Finally we expressed the objective function of minimum variance model using a discrete time linear quadratic regulator and employed Riccati recursion to obtain the optimal movement trajectories of the human forearm. The results of example simulation show that the optimal movement trajectory of the forearm follows a smooth curve, and the speed curve of the hand is single peaked and bell shaped. These are in good agreement with the inherent kinematic properties of optimal movement, and therefore the method is effective for calculating the optimal movement trajectory of the human forearm.

scholarly journals Optimization of the linear quadratic regulator (LQR) control quarter car suspension system using genetic algorithm

2016 ◽  
Vol 36 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Mahesh Nagarkar ◽  
G. J. Vikhe Patil
Keyword(s):  
Genetic Algorithm ◽  
Linear Quadratic Regulator ◽  
Suspension System ◽  
Optimization Approach ◽  
Multi Objective Optimization ◽  
Linear Quadratic ◽  
Multi Objective ◽  
Weighting Matrix ◽  
Car Suspension ◽  
Quarter Car

<p>In this paper, a genetic algorithm (GA) based in an optimization approach is presented in order to search the optimum weighting matrix parameters of a linear quadratic regulator (LQR). A Macpherson strut quarter car suspension system is implemented for ride control application. Initially, the GA is implemented with the objective of minimizing root mean square (RMS) controller force. For single objective optimization, RMS controller force is reduced by 20.42% with slight increase in RMS sprung mass acceleration. Trade-off is observed between controller force and sprung mass acceleration. Further, an analysis is extended to multi-objective optimization with objectives such as minimization of RMS controller force and RMS sprung mass acceleration and minimization of RMS controller force, RMS sprung mass acceleration and suspension space deflection. For multi-objective optimization, Pareto-front gives flexibility in order to choose the optimum solution as per designer’s need.</p>

scholarly journals Actuator Placement for Active Surge Control in a Multi-Stage Axial Compressor

1996 ◽  
Author(s):  
M. Montazeri-Gh. ◽  
D. J. Allerton ◽  
R. L. Elder
Keyword(s):  
Cost Function ◽  
Mass Flow ◽  
Controller Design ◽  
Axial Compressor ◽  
Minimum Cost ◽  
Linear Quadratic Regulator ◽  
Algebraic Riccati Equation ◽  
Linear Quadratic ◽  
Non Linear ◽  
Actuator Placement

This paper describes an actuator placement methodology for the active control of purely one-dimensional instabilities of a seven-stage axial compressor using an air bleeding strategy. In this theoretical study, using stage-by-stage non-linear modelling based on the conservation equations of mass, momentum, and energy, a scheduling LQR (Linear Quadratic Regulator) controller is designed for several actuator locations in a compressor from the first stage to the plenum. In this controller design, the LQR weighting matrices are selected so that the associated cost function includes only air bleeding mass flow leading to the minimisation of the air bleed. The LQR cost function represents a measure of the consumption of air bleeding and can be calculated analytically using the solution of an Algebraic Riccati Equation. From analysis of the cost at different compressor stages, the location of an air bleeding actuator is selected at the stage with the minimum cost. Finally, using an ACSL simulation program, the scheduling controller has been integrated with a non-linear. stage-by-stage model and the time response of the air bleeding mass flow at different locations has been obtained to confirm the results from the analytical approach. Results are presented to show actively stabilised compressor flow beyond the surge point where the air bleed is minimised. These results also indicate the preferred location of the actuator at the compressor downstream stages for both low and high compressor speeds.

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