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.

Robustness of Minimum Norm Quadratic Unbiased Estimator of Variance Under the General Linear Model

Necessary and sufficient conditions for equalities between a 2 y′(I-P Xx)y and minimum norm quadratic unbiased estimator of variance under the general linear model, where a 2 is a known positive number, are derived. Further, when the Gauss? Markov estimators and the ordinary least squares estimator are identical, a relative simply equivalent condition is obtained. At last, this condition is applied to an interesting example.

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.

采用转角补偿LQR的自动驾驶集卡路径跟踪控制

为了解决自动驾驶集卡中半挂车在行驶过程中的路径偏移问题,提出了一种引入铰接角偏差和转角补偿的线性二次调节器(LQR)控制方法。基于集卡三自由度动力学模型建立了考虑铰接角偏差的路径跟踪误差模型;通过采用比例-积分-微分(PID)算法进行转角补偿设计了LQR路径跟踪控制器;通过MATLAB/Simulink和TruckSim搭建联合仿真平台,在不同工况下进行仿真分析验证。结果表明:通过采用提出的路径跟踪算法,引入PID转角补偿,牵引车平均距离偏差降低62%以上,半挂车平均距离偏差降低31%以上,航向偏差和铰接角偏差也均有所改善。因此,该文提出的控制算法具有较好的路径跟踪性能,提高了对期望路径跟踪的精度和稳定性。

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.

Quadratic Stabilization of Switched Uncertain Linear Systems: A Convex Combination Approach

We consider quadratic stabilization for a class of switched systems which are composed of a finite set of continuoustime linear subsystems with norm bounded uncertainties. Under the assumption that there is no single quadratically stable subsystem, if a convex combination of subsystems is quadratically stable, then we propose a state-dependent switching law, based on the convex combination of subsystems, such that the entire switched linear system is quadratically stable. When the state information is not available, we extend the discussion to designing an outputdependent switching law by constructing a robust Luenberger observer for each subsystem.

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.

Singular linear quadratic performance with the worst disturbance rejection for descriptor systems

This paper deals with the problem of singular linear quadratic performance with the worst-disturbance rejection for descriptor systems. Under the conditions we give, the worst-disturbance and the optimal control-state pair are unique respectively, the optimal control can be synthesized as state feedback and the closed-loop system is regular, stable and impulse-free.

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.

Improved Balas and Mazzola Linearization for Quadratic 0-1 Programs with Application in a New CuttingPlane Algorithm

Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed.

All Admissible Linear Estimators under Quadratic Loss in Multivariate Model

For multivariate linear model Y=XΘ+ε, ~N(0, σ 2ΣV), this paper is concerned with the admissibility of linear estimators of estimable function SXΘ in the class of all estimators. All admissible linear estimators of SXΘ are given under each of four definitions of admissibility.

Computing of LQR Technique for Nonlinear System Using Local Approximation

The main idea behind the present research is to design a state-feedback controller for an underactuated nonlinear rotary inverted pendulum module by employing the linear quadratic regulator(LQR)technique using local approximation.The LQR is an excellent method for developing a controller for nonlinear systems.It provides optimal feedback to make the closed-loop system robust and stable,rejecting external disturbances.Model-based optimal controller for a nonlinear system such as a rotatory inverted pendulum has not been designed and implemented using Newton-Euler,Lagrange method,and local approximation.Therefore,implementing LQR to an underactuated nonlinear system was vital to design a stable controller.A mathematical model has been developed for the controller design by utilizing the Newton-Euler,Lagrange method.The nonlinear model has been linearized around an equilibrium point.Linear and nonlinear models have been compared to find the range in which linear and nonlinear models’behaviour is similar.MATLAB LQR function and system dynamics have been used to estimate the controller parameters.For the performance evaluation of the designed controller,Simulink has been used.Linear and nonlinear models have been simulated along with the designed controller.Simulations have been performed for the designed controller over the linear and nonlinear system under different conditions through varying system variables.The results show that the system is stable and robust enough to act against external disturbances.The controller maintains the rotary inverted pendulum in an upright position and rejects disruptions like falling under gravitational force or any external disturbance by adjusting the rotation of the horizontal link in both linear and nonlinear environments in a specific range.The controller has been practically designed and implemented.It is vivid from the results that the controller is robust enough to reject the disturbances in milliseconds and keeps the pendulum arm deflection angle to zero degrees.

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.

GLOBAL LINEAR AND QUADRATIC ONE-STEP SMOOTHING NEWTON METHOD FOR VERTICAL LINEAR COMPLEMENTARITY PROBLEMS

A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).

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.

Simultaneous quadratic performance stabilization for linear time-delay systems

A newly designed approach of simultaneous stabilization is given for linear discrete time-delay systems. The problem of stabilization for a collection of systems is discussed initially. Adequate condition are obtained in terms of linear matrix inequalities （LMIs） which are independent of time delays such that the resultant collection of discrete time-delay systems are stable with an upper bound of the quadratic performance index. Subsequently, controllers are designed such that the resultant closed-loop discrete time-delay systems are simultaneously stabilized with the upper bound of the quadratic performance index. Finally,a numerical example is given to illustrate the design method.

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrixcan be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.

基于LQR的三有源桥变换器控制

针对三有源桥变换器解耦控制中工况远离稳态点解耦矩阵变化时控制性能较差的问题,提出一种基于线性二次型调节器(LQR)的三有源桥变换器控制方法。通过电路等效变换和端口功率传输方程建立状态空间模型,进一步设计基于状态反馈的LQR控制方法,在实现功率潮流控制的同时保持端口电压的稳定,通过仿真控制验证所提出方法的控制性能,对比传统PI解耦控制,在工况变化时,该控制方法具有更好的动态性能。

Kalman Filter and H_(∞)Filter Based Linear Quadratic Regulator for Furuta Pendulum

This paper deals with Furuta Pendulum(FP)or Rotary Inverted Pendulum(RIP),which is an under-actuated non-minimum unstable non-linear process.The process considered along with uncertainties which are unmodelled and analyses the performance of Linear Quadratic Regulator(LQR)with Kalman filter and H∞filter as two filter configurations.The LQR is a technique for developing practical feedback,in addition the desired x shows the vector of desirable states and is used as the external input to the closed-loop system.The effectiveness of the two filters in FP or RIP are measured and contrasted with rise time,peak time,settling time and maximum peak overshoot for time domain performance.The filters are also tested with gain margin,phase margin,disk stability margins for frequency domain performance and worst case stability margins for performance due to uncertainties.The H-infinity filter reduces the estimate error to a minimum,making it resilient in the worst case than the standard Kalman filter.Further,when theβrestriction value lowers,the H∞filter becomes more robust.The worst case gain performance is also focused for the two filter configurations and tested where H∞filter is found to outperform towards robust stability and performance.Also the switchover between the two filters is dependent upon a user-specified co-efficient that gives the flexibility in the design of non-linear systems.The non-linear process is tested for set point tracking,disturbance rejection,un-modelled noise dynamics and uncertainties,which records robust performance towards stability.