Linear Quadratic Regulator Synthesis Research Articles

Combined Observer-Based State Feedback and Optimized P/PI Control for a Robust Operation of Quadrotors

This paper deals with a discrete-time observer-based state feedback control design by taking into consideration bounded parameter uncertainty, actuator faults, and stochastic noise in an inner control loop which is extended in a cascaded manner by outer PI- and P-control loops for velocity and position regulation. The aim of the corresponding subdivision of the quadrotor model is the treatment of the control design in a systematic manner. In the inner loop, linear matrix inequality techniques are employed for the placement of poles into a desired area within the complex z-plane. A robustification of the design towards noise is achieved by optimizing both control and observer gains simultaneously guaranteeing stability in a predefined bounded state domain. This procedure helps to reduce the sensitivity of the inner control loop towards changes induced by the outer one. Finally, a model-based optimization process is employed to tune the parameters of the outer P/PI controllers. To allow for the validation of accurate trajectory tracking, a comparison of the novel approach with the use of a standard extended Kalman filter-based linear-quadratic regulator synthesis is presented to demonstrate the superiority of the new design.

Metaheuristics-based Multi-objective Design of Global Robust Optimal Sliding Mode Control of Discrete Uncertain Systems

The paper introduces a new design control strategy for finite time stabilization of uncertain discrete systems. The proposed control method uses the First Order Sliding Mode Control (FOSMC), the Linear Quadratic Regulator (LQR) and Genetic Algorithms (GA). The key idea of combining two control strategies lies in the fact that a robust controller tackles the uncertainties when the optimal controller performances are unaffected. Firstly, as the performances of the Sliding Mode Control (SMC) greatly depends on the choice of the sliding surface, a novel method based on the resolution of a Sylvester equation is proposed. Secondly, to ensure the robustness in the whole space, the Integral Sliding Mode Control (ISMC) is combined with the above LQR method. Rigorous stability analysis for the closed-loop system is given and proves that the system states are always bounded even the unknown disturbances. In order to handle the problem accompanying the LQR synthesis as well as the gain of the robust compensator, a multi-objective optimization problem considering a set of competing objectives is solved using GA. For that, a novel dynamically aggregated objective function is proposed. A set of non-dominated optimal solutions are provided to the control engineer and then may select the most adequate solution among the resulting optimal Pareto set. The novel method has successfully solved the robust stabilization of two known benchmarks and the different results asserts not merely the efficiency as well as the quality of the designed controller but also its supremacy compared to some existing ones.

Multi-Objective Design of Optimal Sliding Mode Control for Trajectory Tracking of SCARA Robot Based on Genetic Algorithm

This paper focuses on robust optimal sliding mode control (SMC) law for uncertain discrete robotic systems, which are known by their highly nonlinearities, unmodeled dynamics, and uncertainties. The main results of this paper are divided into three phases. In the first phase, in order to design an optimal control law, based on the linear quadratic regulator (LQR), the robotic system is described as a linear time-varying (LTV) model. In the second phase, as the performances of the SMC greatly depend on the choice of the sliding surface, a novel method based on the resolution of a Sylvester equation is proposed. The compensation of both disturbances and uncertainties is ensured by the integral sliding mode control. Finally, to solve the problem accompanying the LQR synthesis, genetic algorithm (GA) is used as an offline tool to search the two weighting matrices. The main contribution of this paper is to consider a multi-objective optimization problem, which aims to minimize not only the chattering phenomenon but also other control performances. A novel dynamically aggregated objective function is proposed in such a way that the designer is provided, once the optimization is achieved, by a set of nondominated solutions and then he selects the most preferable alternative. To show the performance of the new controller, a selective compliance assembly robot arm robot (SCARA) is considered. The results show that the manipulator tracing performance is considerably improved with the proposed control scheme.

Temporally and spatially flexible plan execution for dynamic hybrid systems

Planners developed in the Artificial Intelligence community assume that tasks in the task plans they generate will be executed predictably and reliably. This assumption provides a useful abstraction in that it lets the task planners focus on what tasks should be done, while lower-level motion planners and controllers take care of the details of how the task should be performed. While this assumption is useful in many domains, it becomes problematic when controlling physically embedded systems, where there are often delays, disturbances, and failures. The task plans do not provide enough information about allowed flexibility in task duration and hybrid state evolution. Such flexibility could be useful when deciding how to react to disturbances. An important domain where this gap has caused problems is robotics, particularly, the operation of robots in unstructured, uncertain environments. Due to the complexity of this domain, the demands of tasks to be performed, and the actuation limits of robots, knowledge about permitted flexibility in execution of a task is crucial. We address this gap through two key innovations. First, we specify a Qualitative State Plan (QSP), which supports representation of spatial and temporal flexibility with respect to tasks. Second, we extend compilation approaches developed for temporally flexible execution of discrete activity plans to work with hybrid discrete/continuous systems using a recently developed Linear Quadratic Regulator synthesis algorithm, which performs a state reachability analysis to prune infeasible trajectories, and which determines optimal control policies for feasible state regions. The resulting Model-based Executive is able to take advantage of spatial and temporal flexibility in a QSP to improve handling of disturbances. Note that in this work, we focus on execution of QSPs, and defer the problem of how they are generated. We believe the latter could be accomplished through extensions to existing task planners.

Linear quadratic regulator control of multi‐agent systems

SummaryThis paper considers a collection of agents performing a shared task making use of relative information communicated over an information network. The designed suboptimal controllers are state feedback and static output feedback, which are guaranteed to provide a certain level of performance in terms of a linear quadratic regulator (LQR) cost. Because of the convexity of the LQR performance region, the suboptimal LQR control problem with state feedback is reduced to the solution of two inequalities, with the minimum and maximum eigenvalues of the Laplacian matrix as the coefficients. The advantage of the method is that the LQR control problem of network multi‐agent systems can be converted into the LQR control of a set of single‐agent systems, and the structure constraint on the feedback gain matrix can be eliminated. It can be shown that the size of the LQR control problem will not increase according to the number of the node in the fairly general framework. The method can be extended to the synthesis of the static output feedback, which is derived from the weighting matrices in LQR. Through some coordinate transformation and the augmentation of the output matrix, the LQR synthesis is provided on the basis of the output measurements of the adjacent agents. Numerical examples are presented to illustrate the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Transmission Power Control for Opportunistic QoS Provision in Wireless Networks

This paper proposes an automatic control framework to address the problem of distributed and opportunistic transmission power control in wireless communication networks. The proposed framework allows high flexibility on the quality of service (QoS) provision by exploiting the link quality variation. It associates a standard target tracking algorithm with a dynamic target QoS determined by a linear quadratic regulator (LQR) controller that induces an opportunistic behavior. Strict QoS requirements can also be fulfilled by choosing appropriately the weighting parameters of the LQR synthesis. To prevent performance degradation due to measurement uncertainties, robust solutions are developed. The proposed mixed H2/H∞ distributed power control combines the optimal H2 control and the robust H∞ control to obtain trade-off solutions. All developed algorithms present low computational complexity, and the mixed H2/H∞ distributed power control can operate as the pure H2 control if the robust constraint is relaxed according to the choice of a single parameter. The performance of proposed algorithms is assessed through computer simulations.

Use of Rotor State Feedback to Improve Closed-Loop Stability and Handling Qualities

A rotorcraft control law that uses rotor state feedback (RSF) is presented and demonstrated in simulation. The baseline control law uses a model following/dynamic inversion approach to control the roll, pitch, and yaw axes. The RSF control law was designed to integrate seamlessly with the baseline control law and can be readily engaged or disengaged. The RSF control gains were designed using linear quadratic regulator synthesis. Linear analyses showed that RSF could allow for the feedback gains on rates and attitude to be increased to values that would result in closed-loop instability without the use of RSF. The increased gains can be used to increase bandwidth and improve disturbance rejection. The controller was tested on a nonlinear model in both non–real-time and piloted simulations, and results confirmed the linear analysis. The RSF control law design has potential to improve handling qualities by allowing higher bandwidth and better disturbance rejection with reduced risk of closed-loop instability.

LQ-boundary control of a diffusion-convection-reaction system

In this work, the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations (PDEs) with spatially varying coefficients is studied. An infinite-dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model. The evolutionary model which incorporates the spatially varying coefficients of the underlying set of the PDEs is used for subsequent linear quadratic regulator synthesis. The formulated linear quadratic-state feedback controller is applied to a nonlinear model of the reactor and its performance is studied.

Linear quadratic optimal boundary control of a diffusion-convection-reaction system

AbstractIn this work, the boundary control of a distributed parameter system (DPS) modeled by parabolic partial differential equations with spatially varying coefficients is studied. An infinite dimensional state space setting is formulated and an exact transformation of the boundary actuation is realized to obtain an evolutionary model. The evolutionary model is used for subsequent linear quadratic regulator synthesis which incorporates the spatially varying coefficients of the underlying set of the PDEs. The formulated LQR controller is applied to the nonlinear model of the system and its performance is studied.

Synthesis of robust discrete guidance system for solid body with incomplete state vector measurements

The paper is deals system synthesis method of the robust control. The method is based on estimation of perfo-?mance and robustness represented by Н2 - and Н - norms accordingly. Particularities of the control object (solid body) establish the following sequence of steps in order to solve a problem: 1) restoring the full state-space vector applying the Luenberger observer; 2) optimal linear-quadratic regulator synthesis; 3) utilization the procedure of Н2/Н robust optimization. This technique has been applied taking into account the constraints on oscillations of the closed-loop system. The same procedure has been applied without the above-mentioned constraints. Simulation results have confirmed the effectiveness and efficiency of the proposed approach.

Linear quadratic optimal output feedback control for systems with poles in a specified region

The problem of output feedback linear quadratic regulator synthesis with regional pole constraints is studied. The synthesis algorithm is presented and its convergence is also proved. The effectiveness of the algorithm is illustrated by an example.

Control issues of microgravity vibration isolation

Active vibration isolation systems contemplated for microgravity space experiments may be designed to reach given performance requirements in a variety of ways. An analogy to passive isolation systems proves to be illustrative but lacks the flexibility as a design tool of a control systems approach and may lead to poor designs. For example, it is shown that a focus on equivalent stiffness in isolation system design leads to a controller that sacrifices robustness for performance. Control theory as applied to vibration isolation is reviewed and passive analogies are discussed. The loop shaping trade-off is introduced and used to design a single degree of freedom feedback controller. An algebraic control design methodology is contrasted to loop shaping and critiqued. Multi-axis vibration isolation and the problems of decoupled single loop control are introduced through a two degree of freedom example problem. It is shown that center of mass uncertainty may result in instability when decoupled single loop control is used. This results from the ill conditioned nature of the feedback control design. The use of the Linear Quadratic Regulator synthesis procedure for vibration isolation controller design is discussed.

Use of negative weights in linear quadratic regulator synthesis

An inverse problem of optimal linear quadratic regulators (LQR) is examined for single-input systems, and the selection of the weighting matrices that achieve a specified pole location is also discussed. In particular, the Kessler polynomial is used as a desirable pole location, and the weighting matrices are derived in an analytical form. Although this pole specification results in the use of some negative diagonal weights in the performance index, the existence and uniqueness of the Riccati solution are guaranteed by Molinari's theorem. Through the sacrifice of the circle condition, it is shown that some of the deficiencies of the LQR controllers are avoided, and that several characteristics (which classical controllers provide, but which modern methods cannot) are retained. An application to roll autopilot systems for missiles is given to illustrate and substantiate the proposed method, as well as to compare it with the conventional LQR.

Periodic-disturbance accommodating control of the Space Station for asymptotic momentum management

Periodic-disturbance accommodating control is investigated for asymptotic momentum management of control moment gyros used as primary actuating devices for the Space Station. The proposed controller utilizes the concepts of quaternion feedback control and periodic-disturbance accommodation to achieve oscillations about the constant torque equilibrium attitude, while minimizing the control effort required. Three-axis coupled equations of motion, written in terms of quaternions, are derived for roll/yaw controller design and stability analysis. The quaternion feedback controller designed using the linear-quadratic regulator synthesis technique is shown to be robust for a wide range of pitch angles. It is also shown that the proposed controller tunes the open-loop unstable vehicle to a stable oscillatory motion which minimizes the control effort needed for steady-state operations.

Linear-quadratic-regulator pointing control system for a high-altitude balloon payload

A pointing control system design for the science package of a NASA high-altitude research balloon is described. The balloon assembly consists of a single helium balloon connected to a payload recovery parachute, payload gondola, and ballast hopper. Pointing of the scientific payload is accomplished via an arrangement of drive motors and a flywheel. Linear quadratic regulator (LQR) synthesis techniques are employed to produce the azimuth and elevation controller designs. The use of LQR synthesis is motivated by the azimuthal dynamic coupling encountered between the balloon and gondola. Two control devices are employed in azimuth, one of which is a decoupler motor and the other a flywheel. The decoupler motor is intended to isolate the gondola from the balloon such that the flywheel can be accelerated or decelerated about a steady-state angular velocity to provide precise azimuthal pointing. The multiple-input/multiple-output nature of the azimuth pointing problem is best handled in a matrix synthesis procedure such as LQR. The controller design methodology is explained, and a combination of time responses and singular value analyses are used to analytically evaluate the performance of the control system. 11 refs., 17 figs.