Linear Quadratic Regulator Design Research Articles

Reduction of data amount in data-driven design of linear quadratic regulators

Reduction of data amount in data-driven design of linear quadratic regulators

LQR and Fuzzy-PID Control Design on Double Inverted Pendulum

Double inverted pendulum is a non-linear and unstable system. Double inverted pendulum can be stabilized in the upright position by providing control to the system. In this research we compare two types of controllers namely Linear Quadratic Regulator (LQR) and Fuzzy-PID. The objective is to determine the control strategy that provides better performance on the position of the cart and pendulum angle. We modelled the system which is then linearized and given control. From the simulation results, it is proven that LQR and Fuzzy-PID controllers have been successfully designed to stabilize the double inverted pendulum. However, when given a disturbance in the form of noise step, the LQR controller has not been able to achieve the desired reference for up to 20 seconds. In another hand, the Fuzzy-PID controller is able to achieve the desired reference after 8 seconds. Therefore, it can be concluded that the Fuzzy-PID controller when applied to the Double Inverted pendulum system has better performance than the LQR controller.

Comparison of optimization approaches on linear quadratic regulator design for trajectory tracking of a quadrotor

Comparison of optimization approaches on linear quadratic regulator design for trajectory tracking of a quadrotor

Data-driven model reduction approach for active vibration control of cable-strut structures

The cable-strut structures have the features of low stiffness and weak damping, resulting in large vibration response under dynamic excitation. The vibration response of the cable-strut structures can be reduced by active control methods. However, many traditional active control methods need to build a state-space model (SSM) based on finite element model (FEM), and solving the active control force for the high-dimensional SSM is time-consuming. In this paper, the dynamic mode decomposition with control (DMDc) method is proposed to achieve a data-driven reduced-order SSM for active vibration control of cable-strut structures. This proposed method only uses the data of structural response and control force input to build reduced-order SSM and does not rely on the FEM. Linear quadratic regulator (LQR) design using the DMDc-based reduced-order SSM (DMDc-LQR) is more computationally efficient than that using the FEM-based SSM (FEM-LQR). The relationship between the design parameters of the DMDc-LQR controller and the FEM-LQR controller is established, which ensures that the control performance indexes of the DMDc-LQR controller and the FEM-LQR controller are equal. The effectiveness of the proposed DMDc-LQR method is validated using numerical Kiewitt cable dome and tensegrity grid under wind and seismic loads. The results show that the DMDc-LQR controller maintains similar control effect to that of the FEM-LQR controller and has advantage of high computing efficiency as well. In addition, the influence of critical parameters (i.e., data noise and model order) on vibration control effect is comprehensively explored. It is found that increasing the model order is effective to mitigate the noise influence on the DMDc-based reduced-order SSM and ensures the effectiveness of the DMDc-LQR controller.

Robust output-feedback control of fluid resuscitation with absolute stability guarantee against uncertain physiology and therapeutic responsiveness

This paper presents a robust closed-loop control approach to fluid resuscitation in patients with hemorrhagic blood loss. A unique strength of the proposed approach is its robustness against uncertain and time-varying patient physiology and therapeutic effectiveness. First, we adopted an observer-based control architecture that can fulfill set point tracking and disturbance rejection objectives. Second, we determined the control gains to achieve adequate transient response performance using the linear quadratic regulator design. Third, we determined the observer gains as a solution to a set of linear matrix inequalities so that the overall closed-loop fluid resuscitation control system is (i) robust against the variability in patient physiology and (ii) absolutely stable against unknown therapeutic effectiveness. We demonstrated the initial proof-of-concept of the proposed approach by conducting rigorous in silico testing using a large number of physiologically plausible virtual patients, while ascertaining the absolute stability via the circle criterion analysis. The results suggested that the proposed approach to closed-loop control of fluid resuscitation is a promising option to advance automation of fluid resuscitation armed with stability against a large variability in patient physiology and therapeutic effectiveness as well as adequate performance in set point tracking and disturbance rejection.

Optimal analytical cascade control for FOPDT systems

This paper proposes a novel control strategy based on optimal Linear Quadratic Regulator (LQR) design applied to cascade systems with stable, integrative, and unstable First Order plus Dead-Time (FOPDT) outer loop models. The secondary (inner) and primary (outer) loop controller designs are performed using the Simplified Filtered Smith Predictor (SFSP). The optimal analytical design is derived from a novel strategy where the pole of the internal loop is placed so that the LQR criterion can be achieved with a simple tuning of the external loop. In contrast to related work, the secondary controller is designed using an SFSP control structure to nominally achieve optimal LQR performance and accelerated rejection of disturbances acting on the inner-loop process. To illustrate the usefulness of the proposed design, we evaluated the strategy in different scenarios, showing the optimality and performance of the analytical design.

Improvement of the linear quadratic regulator control applied to a DC-DC boost converter driving a permanent magnet direct current motor

This article discusses a new robust control technique that enables the DC-DC boost converter driving a permanent magnet direct current (PMDC) motor to operate in high static and dynamic performances. The new technique is based on the design of a both linear quadratic regulator (LQR) and linear quadratic regulator-proportional integral (LQR-PI) type controllers, which have the advantage of eliminating oscillations, overshoots and fluctuations on different characteristics in steady-state system operation. In order to increase the output voltage, the LQR regulator is combined with a first-order system represented in the form of a closed-loop transfer function, the latter raising the output voltage to 24 volts, this voltage is enough to drive the permanent magnet direct current motor. The contribution of this paper is the creation of a robust control system represented in the form of a hybrid corrector able to regulate steady-state and transient disturbances and oscillations as well as to increase DC-DC boost converter output voltage for the PMDC motor to operate at rated voltage. The results of the three control techniques are validated by MATLAB Simulink.

Novel Results on Output-Feedback LQR Design

This paper provides novel developments in output-feedback stabilization for linear time-invariant systems within the linear quadratic regulator (LQR) framework. First, we derive the necessary and sufficient conditions for output-feedback stabilizability in connection with the LQR framework. Then, we propose a novel iterative Newton's method for output-feedback LQR design and a computationally efficient modified approach that requires solving only a Lyapunov equation at each iteration step. We show that the proposed modified approach guarantees convergence from a stabilizing state-feedback to a stabilizing output-feedback solution and succeeds in solving high dimensional problems where other, state-of-the-art methods, fail. Finally, numerical examples illustrate the effectiveness of the proposed methods.

Design of Linear Quadratic Regulator Based on Genetic Model Reference Adaptive Control

As the conventional control system is a controller that controls or regulates the dynamics of any other process. From time to time, this control system may not behave appropriately online; this is because of many factors like some variation in the dynamics of the process itself or unexpected changes in the environment, or even undefined parameters of the system model. To overcome the problem above, an adaptive controller is designed and implemented. This paper discusses the design of a controller for a ball and beam system with Genetic Model Reference Adaptive Control (GMRAC) for adaptive mechanism with the MIT rule. Parameter adjustment (selection) should be done using optimization methods in order to obtain an optimal performance, so genetic algorithm. (GA) will be used as an optimization method to obtain the optimum values for these parameters. As the controller, a Linear quadratic regulator (LQR) controller will be used as it is one of the most popular controllers, the performance of the proposed controller with the ball and beam system will be carried out with MATLAB Simulink in order to evaluate its effectiveness. The results show satisfactory performance where the position of the ball tracks the desired model reference.

Linear Quadratic Control of Positive Systems: A Projection-Based Approach

This paper investigates the nonfragile design of linear quadratic regulator (LQR) for continuous-time positive linear systems subject to gain variations. Based on positive systems theory and Lyapunov theory, the solvability and optimality of the positivity-preserving LQR problem are analyzed through the lens of optimization, and two projection theorems are derived for single-input and multi-input positive systems, respectively, which paves the way for developing the projected gradient descent (PGD) algorithm for computation. The proposed results fill the literature gap by considering both the optimality and positivity in LQR design, and finally numerical simulations are provided to verify the effectiveness of the results.

Complex Fractional-Order LQIR for Inverted-Pendulum-Type Robotic Mechanisms: Design and Experimental Validation

This article presents a systematic approach to formulate and experimentally validate a novel Complex Fractional Order (CFO) Linear Quadratic Integral Regulator (LQIR) design to enhance the robustness of inverted-pendulum-type robotic mechanisms against bounded exogenous disturbances. The CFO controllers, an enhanced variant of the conventional fractional-order controllers, are realised by assigning pre-calibrated complex numbers to the order of the integral and differential operators in the control law. This arrangement significantly improves the structural flexibility of the control law, and hence, subsequently strengthens its robustness against the parametric uncertainties and nonlinear disturbances encountered by the aforementioned under-actuated system. The proposed control procedure uses the ubiquitous LQIR as the baseline controller that is augmented with CFO differential and integral operators. The fractional complex orders in LQIR are calibrated offline by minimising an objective function that aims at attenuating the position-regulation error while economising the control activity. The effectiveness of the CFO-LQIR is benchmarked against its integer and fractional-order counterparts. The ability of each controller to mitigate the disturbances in inverted-pendulum-type robotic systems is rigorously tested by conducting real-time experiments on Quanser single-link rotary pendulum system. The experimental outcomes validate the superior disturbance rejection capability of the CFO-LQIR by yielding rapid transits and strong damping against disturbances while preserving the control input economy and closed-loop stability of the system.

Robust Optimal Output-Tracking Control of Constrained Mechanical Systems With Application to Autonomous Rovers

This article presents a robust optimal output-tracking control strategy for underactuated mechanical systems whose motion is restricted by mixed holonomic and nonholonomic constraints. Autonomous rovers/cars and unmanned underwater/aerial vehicles are a few examples of such systems that must often operate in harsh environments and under uncertain conditions. We present a comprehensive control analysis of this large class of nonlinear systems, including the existing studies on local reachability and static state feedback linearization. We also propose a local observability analysis of the feedback transformed input–output linearized systems. Based on the input–output linearization of the holonomically restricted nominal system, we develop a sliding mode control strategy that is robust against the projected effects of uncertainties and disturbances on the system’s output. Asymptotic stability of the output toward a bounded desired trajectory is proven using Lyapunov’s direct method, while the system’s internal stability (in the sense of boundedness) is investigated based on the notion of tracking-error zero dynamics. Time-dependent bounded matched uncertainties in the inertia parameters and disturbance forces arising in the unrestricted system are considered in this study. We propose an optimal sliding manifold according to the finite-horizon linear-quadratic regulator design problem with split boundary value conditions. The developed control strategy is implemented on a six-wheel autonomous Lunar rover in a simulation environment and its performance is compared with that of an optimal proportional–integral–derivative feedback, feedforward controller. The optimal sliding mode controller shows superior performance in trajectory tracking with acceptable control actions.

Linear quadratic regulator design via metaheuristics applied to the damping of low-frequency oscillations in power systems

This paper proposes the application of control by state feedback using the linear quadratic regulator (LQR) optimized by metaheuristics to damp low-frequency electromechanical oscillations in electrical power systems. The current sensitivity model was used to represent the single machine infinite bus (SMIB) system in the time domain. The weighting matrices of the LQR were adjusted using four different algorithms: (i) the genetic algorithm, (ii) the differential evolution algorithm, (iii) the particle swarm optimization algorithm, and (iv) the gray wolf optimization (GWO) algorithm. In the cases considered, disturbances were applied to the electrical power system and, then, performances comparisons associated with each metaheuristic were statistically analyzed, in which the number of iterations, error, and time to achieve convergence of each algorithm were compared. From the results, it was possible to conclude that the algorithms were efficient in adjusting the weighting matrices of the LQR, providing additional damping to the poles of interest of the system. It was also possible to conclude that the GWO algorithm presented the best performance, accrediting it as a powerful tool in the study of small-signal stability for the analyzed case.

Sub-Optimal Stabilizers of the Pendubot Using Various State Space Representations

This paper considers the issue of linear-quadratic regulator (LQR) design for nonlinear systems with the use of smooth state and input transformations. The proposed design methodology is considered in the stabilisation task of the Pendubot, which is based on the concept of feedback equivalent control systems. It turns out that it is possible to find a controller that ensures comparable dynamics of the closed-loop system in the vicinity of the set point regardless of the state-space representation adopted. In addition, the synthesis of suboptimal controllers according to the LQR strategy ensuring equal dynamics at the equilibrium point is presented. The properties of the studied controllers were investigated in a simulation environment and using experimental tests. The detailed forms of transformations and linear approximations given can be regarded as ready-made procedures that can be applied to stabilise similar mechanical systems in robotics.

Weighted Linearization of Nonlinear Systems

This brief proposes a generalization of the linearization technique in which the computation of the Jacobian matrices at the state trajectory of interest is replaced by the multiple integral over the state and input spaces of the Jacobian matrix functions multiplied by a weighting function. The standard linearization can be recovered as a particular case of the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weighted linearization</i> by choosing the weighting function as an appropriately shifted multivariable Dirac delta function. Linear-quadratic regulator (LQR) design is used in the illustrative examples to show the potentiality of the proposed technique.

Optimization of automobile active suspension system using minimal order

&lt;p&gt;&lt;span&gt;This paper presents an analysis and design of linear quadratic regulator for reduced order full car suspension model incorporating the dynamics of the actuator to improve system performance, aims at benefiting: Ride comfort, long life of vehicle, and stability of vehicle. Vehicle’s road holding or handling and braking for good active safety and driving pleasure, and keeping vehicle occupants comfortable and reasonably well isolated from road noise, bumps, and vibrations are become a key research area conducted by many researchers around the globe. Different researchers were tested effectiveness of different controllers for different vehicle model without considering the actuator dynamics. In this paper full vehicle model was reduced to a minimal order using minimal realization technique. The entire system responses were simulated in MATLAB/Simulink environment. The effectiveness of linear quadratic regulator controller was compared for the system model with and without actuator dynamics for different road profiles. The simulation results were indicated that percentage reduction in the peak value of vertical and horizontal velocity for the linear quadratic regulator with actuator dynamics relative to linear quadratic regulator without actuator dynamics was 28.57%. Overall simulation results were demonstrated that proposed control scheme has able to improve the effectiveness of the car model for both ride comfort and stability.&lt;/span&gt;&lt;/p&gt;

Design and Experimental Comparison of PID, LQR and MPC Stabilizing Controllers for Parrot Mambo Mini-Drone

Parrot Mambo mini-drone is a readily available commercial quadrotor platform to understand and analyze the behavior of a quadrotor both in indoor and outdoor applications. This study evaluates the performance of three alternative controllers on a Parrot Mambo mini-drone in an interior environment, including Proportional–Integral–Derivative (PID), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC). To investigate the controllers’ performance, initially, the MATLAB®/Simulink™ environment was considered as the simulation platform. The successful simulation results finally led to the implementation of the controllers in real-time in the Parrot Mambo mini-drone. Here, MPC surpasses PID and LQR in ensuring the system’s stability and robustness in simulation and real-time experiment results. Thus, this work makes a contribution by introducing the impact of MPC on this quadrotor platform, such as system stability and robustness, and showing its efficacy over PID and LQR. All three controllers demonstrate similar tracking performance in simulations and experiments. In steady state, the maximal pitch deviation for the PID controller is 0.075 rad, for the LQR, it is 0.025 rad, and for the MPC, it is 0.04 rad. The maximum pitch deviation for the PID-based controller is 0.3 rad after the take-off impulse, 0.06 rad for the LQR, and 0.17 rad for the MPC.

Discounted-cost linear quadratic regulation of switched linear systems

This paper investigates the design of discounted-cost linear quadratic regulator for a class of switched linear systems. The distinguishing feature of the proposed method is that the designed discounted-cost linear quadratic regulator achieves not only the desired optimisation index but also the exponentially convergent of the state trajectory of the closed-loop switched linear systems. First, the studied problem is transformed into a quadratic-programming problem by embedding transformation. Then, the bang-bang type solution of the embedded optimal control problem on a finite-time horizon is derived, which is the optimal solution to the original problems. The computable sufficient conditions on discounted-cost linear quadratic regulator are proposed for finite-time and infinite-time horizon cases, respectively. Finally, two examples are provided to demonstrate the effectiveness of the proposed method.

On the robustness of networked cooperative tracking systems

This paper provides an analytical framework for the robustness of networked multi-agent systems (MAS). It is well-known that a single-agent linear quadratic regulator (LQR) system can guarantee 60° phase margin and infinite gain margin. However, for networked MAS, there exist no theoretical results on guaranteed stability margins, due to the complexity caused by the interplay of communication structure and agents’ dynamics. In this paper, we analyze the effect of communication graph topology on the robustness properties of networked cooperative tracking systems with local LQR designs. For such systems, we provide closed-form expressions of phase and gain margins modulated by their graph topology, following a Lyapunov type of analysis. We further derive upper bounds of stability margins for MAS with general graph topology, through a structural analysis based on the algebraic graph theory. We prove that the directed tree communication topology is among the most robust graph topology that promises the best stability margins, which are as good as the ones in a single-agent LQR system

Design of a Novel AGC Action for a Linked Hydro Governing System

This paper present an initial effort of collaborating hybrid bacteria foraging oriented particle swarm optimization with linear quadratic regulator (LQR) design for automatic generation control (AGC) of power delivery system having power generation through hydro turbines in interconnected control areas to match the frequency variations as well as tie-power interchange to the well planned value after sudden load disturbance in interconnected system. The enactment of LQR highly depend upon the selection of state cost weighing matrix (Q) and control cost weighting matrix (R) in order to achieve the best control efforts. However, most of the researchers assume the structure of Q and R to be an identity and hence the LQR control action was restricted within certain range. Hence, an early attempt is made to compute the structure of Q and R for LQR through hybrid bacteria foraging oriented PSO for AGC of hydro system and application results are computed for various cases considering error evaluated and graphical AGC responses for step load alteration, influence of AGC non-linearities on the output of AGC, parametric alterations and random load alteration to show the efficacy as well as robustness of the present control design.