Theoretical Study of Double Cost Function Linear Quadratic Regulator(LQR)

Double cost function linear quadratic regulator (DLQR) is developed from LQR theory to solve an optimal control problem with a general nonlinear cost function. In addition to the traditional LQ cost function, another free form cost function was introduced to express the physical need plainly and optimize weights of LQ cost function using the search algorithms. As an instance, DLQR was applied in determining the control input in the front steering angle compensation control (FSAC) model for heavy duty vehicles. The brief simulations show that DLQR is powerful enough to specify the engineering requirements correctly and balance many factors effectively. The concept and applicable field of LQR are expanded by DLQR to optimize the system with a free form cost function.

Robust Fuzzy Tracking Control for Nonlinear Networked Control Systems with Integral Quadratic Constraints

This paper investigates the robust tracking control problcm for a class of nonlinear networked control systems （NCSs） using the Takagi-Sugeno （T-S） fuzzy model approach. Based on a time-varying delay system transformed from the NCSs, an augmented Lyapunov function containing more useful information is constructed. A less conservative sufficient condition is established such that the closed-loop systems stability and time-domain integral quadratic constraints （IQCs） are satisfied while both time-varying network- induced delays and packet losses are taken into account. The fuzzy tracking controllers design scheme is derived in terms of linear matrix inequalities （LMIs） and parallel distributed compensation （PDC）. Furthermore, robust stabilization criterion for nonlinear NCSs is given as an extension of the tracking control result. Finally, numerical simulations are provided to illustrate the effectiveness and merits of the proposed method.

Adaptive Linear Quadratic Regulator for Continuous-Time Systems With Uncertain Dynamics

In this paper, adaptive linear quadratic regulator(LQR) is proposed for continuous-time systems with uncertain dynamics. The dynamic state-feedback controller uses inputoutput data along the system trajectory to continuously adapt and converge to the optimal controller. The result differs from previous results in that the adaptive optimal controller is designed without the knowledge of the system dynamics and an initial stabilizing policy. Further, the controller is updated continuously using input-output data, as opposed to the commonly used switched/intermittent updates which can potentially lead to stability issues. An online state derivative estimator facilitates the design of a model-free controller. Gradient-based update laws are developed for online estimation of the optimal gain. Uniform exponential stability of the closed-loop system is established using the Lyapunov-based analysis, and a simulation example is provided to validate the theoretical contribution.

Linear Quadratic Integral Control for the Active Suspension of Vehicle

The quarter model of an active suspension is established in the form of controllable autoregressive moving average （CARMA） model. An accelerometer can be mounted on the wheel hub for measuring road disturbance; this signal is used to identify the CARMA model parameters by recursive forgetting factors least square method. The linear quadratic integral （LQI） control method for the active suspension is presented. The LQI control algorithm is fit for vehicle suspension control, for the control performance index can comprise multi controlled variables. The simulation results show that the vertical acceleration and suspension travel both are decreased with the LQI control in the low frequency band, and the suspension travel is increased with the LQI control in the middle or high frequency band. The suspension travel is very small in the middle or high frequency band, the suspension bottoming stop will not happen, so the vehicle ride quality can be improved apparently by the LQI control.

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.

One Kind of Fully Coupled Linear Quadratic Stochastic Control Problem with Random Jumps

有随机的一个种有点线性的二次的随机的控制问题跳被学习。最佳的控制的明确的形式被获得。最佳的控制能被证明唯一。一个种概括 Riccati 方程系统被介绍，它的解决之可能性被讨论。为有随机的最佳的控制问题的线性反馈管理者跳被概括 Riccati 方程系统的解决方案给。

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control （LQ control） problems for linear timevarying systems accurately and efficiently, whose key points are solutions of differential Riccati equation （DRE） with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.

A Control Strategy Combining Fuzzy Control and Linear Quadratic Control for Acrobots

A control strategy based on a combination of fuzzy control and linear quadratic control to control the acrobot is presented. The control torque to swing up is directly derived based on the energy of the acrobot. A fuzzy controller is designed to regulate the amplitude of the control torque from the energy during the upswing. After the acrobot enters a neighborhood of the straight up equilibrium position, a linear quadratic regulator is designed to balance it. The proposed control strategy simplifies the control of the acrobot and achieves better performance. The simulation results show the validity of the control strategy.

Global Optimization for Solving Linear Non-Quadratic Optimal Control Problems

This paper presents a global optimization approach to solving linear non-quadratic optimal control problems. The main work is to construct a differential flow for finding a global minimizer of the Hamiltonian function over a Euclid space. With the Pontryagin principle, the optimal control is characterized by a function of the adjoint variable and is obtained by solving a Hamiltonian differential boundary value problem. For computing an optimal control, an algorithm for numerical practice is given with the description of an example.

Spacecraft formation control strategy on Sun-Earth Lissajous orbit

To carry out the deep space exploration tasks near Sun-Earth Libration point L2, the CRTBP dynamic model was built up and the numerical conditional quasi-periodic orbit （Lissajons orbit） was computed near L2. Then, a formation controller was designed with linear matrix inequality to overcome the difficuhy of parameter tuning. To meet the demands of formation accuracy and present thruster＇s capability, a threshold scheme was adopted for formation control. Finally, some numerical simulations and analysis were completed to demonstrate the feasibility of the proposed control strategy.

Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

Formation mechanism and criterion of linear segregation in ZL205A alloy

The experiments and numerical simulation were conducted for ZL205A aluminum alloy cylindrical shell casting. The formation mechanism of the linear segregation produced by the low pressure die casting (LPDC) process was investigated. And the heat transfer of the casting during solidification process was analyzed by simulation technique, resulting from the information of linear segregation obtained by plenty of experiments. The new linear segregation criterion was proposed through the simulation and experimental results. It was found that the melting metal with high Cu contents was feeding the crack shrinkage formed by the tearing under the effect of feeding pressure during the later solidification, which led to the formation of linear segregation. The control methods for the linear segregation were suggested based on the proposed mechanism. Finally, the criterion of linear segregation was confirmed by the production of the actual castings.

Optimal Cooperative Secondary Control for Islanded DC Microgrids via a Fully Actuated Approach

DC-DC converter-based multi-bus DC microgrids(MGs) in series have received much attention, where the conflict between voltage recovery and current balancing has been a hot topic. The lack of models that accurately portray the electrical characteristics of actual MGs while is controller design-friendly has kept the issue active. To this end, this paper establishes a large-signal model containing the comprehensive dynamical behavior of the DC MGs based on the theory of high-order fully actuated systems, and proposes distributed optimal control based on this. The proposed secondary control method can achieve the two goals of voltage recovery and current sharing for multi-bus DC MGs. Additionally, the simple structure of the proposed approach is similar to one based on droop control, which allows this control technique to be easily implemented in a variety of modern microgrids with different configurations. In contrast to existing studies, the process of controller design in this paper is closely tied to the actual dynamics of the MGs. It is a prominent feature that enables engineers to customize the performance metrics of the system. In addition, the analysis of the stability of the closed-loop DC microgrid system, as well as the optimality and consensus of current sharing are given. Finally, a scaled-down solar and battery-based microgrid prototype with maximum power point tracking controller is developed in the laboratory to experimentally test the efficacy of the proposed control method.

Predictive active control of building structures using LQR and artificial intelligence

This study presents a neural network-based model for predicting linear quadratic regulator(LQR)weighting matrices for achieving a target response reduction.Based on the expected weighting matrices,the LQR algorithm is used to determine the various responses of the structure.The responses are determined by numerically analyzing the governing equation of motion using the state-space approach.For training a neural network,four input parameters are considered:the time history of the ground motion,the percentage reduction in lateral displacement,lateral velocity,and lateral acceleration,Output parameters are LQR weighting matrices.To study the effectiveness of an LQR-based neural network(LQRNN),the actual percentage reduction in the responses obtained from using LQRNN is compared with the target percentage reductions.Furthermore,to investigate the efficacy of an active control system using LQRNN,the controlled responses of a system are compared to the corresponding uncontrolled responses.The trained neural network effectively predicts weighting parameters that can provide a percentage reduction in displacement,velocity,and acceleration close to the target percentage reduction.Based on the simulation study,it can be concluded that significant response reductions are observed in the active-controlled system using LQRNN.Moreover,the LQRNN algorithm can replace conventional LQR control with the use of an active control system.

Least Squares Solution for Discrete Time Nonlinear Stochastic Optimal Control Problem with Model-Reality Differences

In this paper, an efficient computational approach is proposed to solve the discrete time nonlinear stochastic optimal control problem. For this purpose, a linear quadratic regulator model, which is a linear dynamical system with the quadratic criterion cost function, is employed. In our approach, the model-based optimal control problem is reformulated into the input-output equations. In this way, the Hankel matrix and the observability matrix are constructed. Further, the sum squares of output error is defined. In these point of views, the least squares optimization problem is introduced, so as the differences between the real output and the model output could be calculated. Applying the first-order derivative to the sum squares of output error, the necessary condition is then derived. After some algebraic manipulations, the optimal control law is produced. By substituting this control policy into the input-output equations, the model output is updated iteratively. For illustration, an example of the direct current and alternating current converter problem is studied. As a result, the model output trajectory of the least squares solution is close to the real output with the smallest sum squares of output error. In conclusion, the efficiency and the accuracy of the approach proposed are highly presented.

Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

This paper presents a contribution related to the control of nonlinear variable-speed marine current turbine(MCT)without pitch operating below the rated marine current speed.Given that the operation of the MCT can be divided into several operating zones on the basis of the marine current speed,the system control objectives are different for each zone.To deal with this issue,we develop a new control approach based on a linear quadratic regulator with variable generator torque.Our proposed approach enables the optimization of the rotational speed of the turbine,which maximizes the power extracted by the MCT and minimizes the transient loads on the drivetrain.The novelty of our study is the use of a real profile of marine current speed from the northern coasts of Morocco.The simulation results obtained using MATLAB Simulink indicate the effectiveness and robustness of the proposed control approach on the electrical and mechanical parameters with the variations of marine current speed.

Guaranteed cost control for saturated uncertain linear systems

In this paper, we discuss the problem of guaranteed cost control for uncertain linear systems subject to actuator saturation. Based on the quadratic and non-quadratic Lyapunov functions, sufficient conditions for the robust stability and performance are derived. Moreover, all the conditions can be expressed as linear matrix inequalities （LMIs） or bilinear matrix inequalities （BMIs） in terms of the feedback gain. Thus, the static controller can be effectively synthesized via convex optimization. A numerical example illustrates the effectiveness of the method.

Containment control for heterogeneous multi-agent systems with time delay via an output regulation approach

This paper presents the containment analysis and design of heterogeneous linear multi-agent systems(MAS)with time-delay under the output regulation.The leaders are treated as exosystems and an modified output regulation error is designed,which can deal with more than one leader in containment control,then the containment problem will be turned into an output regulation problem.A novel analysis framework of the output regulation is proposed to design a dynamic state feedback control law for containment error and distributed observer when the agents cannot receive external system signal,which guarantees the convergence of all follower agents to the dynamic convex hull spanned by the leaders.The system stability for time-delay containment is proved by the output regulation method instead of the Lyapunov method.Finally,a numerical example is given to illustrate the validity of the theoretical results.

General Integral Control Design via Feedback Linearization

Based on the feedback linearization technique, we present a systematic design method for the General Integral Control and a new integral control strategy along with a class of fire-new integrator. By using the linear system theory and Lyapunov method along with LaSalle’s invariance principle, the conditions on the control gains to ensure regionally as well as semi-globally asymptotic stability are provided. Theoretical analysis and simulation results demonstrated that: by using this design method, General Integral Control can deal with nonlinearity and uncertainties of dynamics more effectively;the optimum response can be achieved in the whole control domain, even under uncertain payload and varying-time disturbances. This means that General Integral Control has strong robustness, fast convergence, good flexibility, and then makes the engineers design a high performance controller more easily.

Robust H_∞ control of piecewise-linear chaotic systems with random data loss

This paper studies the problem of robust H∞ control of piecewise-linear chaotic systems with random data loss. The communication links between the plant and the controller are assumed to be imperfect （that is, data loss occurs intermittently, which appears typically in a network environment）. The data loss is modelled as a random process which obeys a Bernoulli distribution. In the face of random data loss, a piecewise controller is designed to robustly stabilize the networked system in the sense of mean square and also achieve a prescribed H∞ disturbance attenuation performance based on a piecewise-quadratic Lyapunov function. The required H∞ controllers can be designed by solving a set of linear matrix inequalities （LMIs）. Chua＇s system is provided to illustrate the usefulness and applicability of the developed theoretical results.