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.

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.

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

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.

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.

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.

An Overview of Sequential Approximation in Topology Optimization of Continuum Structure

This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures.These structures,commonly encountered in engineering applications,often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables.As a result,sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges.Over the past several decades,topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds.In comparison to the large-scale equation solution,sensitivity analysis,graphics post-processing,etc.,the progress of the sequential approximation functions and their corresponding optimizersmake sluggish progress.Researchers,particularly novices,pay special attention to their difficulties with a particular problem.Thus,this paper provides an overview of sequential approximation functions,related literature on topology optimization methods,and their applications.Starting from optimality criteria and sequential linear programming,the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables.In addition,recent advancements have led to the emergence of approaches such as Augmented Lagrange,sequential approximate integer,and non-gradient approximation are also introduced.By highlighting real-world applications and case studies,the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area.Furthermore,to provide a comprehensive overview,this paper offers several novel developments that aim to illuminate potential directions for future research.

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.

Optimal Decoupling Control Method and Its Application to a Ball Mill Coal-pulverizing System

Abstract-The conventional optimal tracking control method cannot realize decoupling control of linear systems with a strong coupling property. To solve this problem, in this paper, an optimal decoupling control method is proposed, which can simultaneousiy provide optimal performance. The optimal decoupling controller is composed of an inner-loop decoupling controller and an outer-loop optimal tracking controller. First, by introducing one virtual control variable, the original differential equation on state is converted to a generalized system on output. Then, by introducing the other virtual control variable, and viewing the coupling terms as the measurable disturbances, the generalized system is open-loop decoupled. Finally, for the decoupled system, the optimal tracking control method is used. It is proved that the decoupling control is optimal for a certain performance index. Simulations on a ball mill coal-pulverizing system are conducted. The results show the effectiveness and superiority of the proposed method as compared with the conventional optimal quadratic tracking （LQT） control method.

Fuzzy Optimal Control Design for Ship Steering

This paper presents a method to design a control scheme for nonlinear systems using fuzzy optimal control.In the design process,the nonlinear system is first converted into local subsystems using sector non linearity approach of Takagi Sugeno(T S)fuzzy modeling.For each local subsystem,an optimal control is designed.Then,the parameters of local controllers are defuzzified to construct a global optimal controller.To prove the effectiveness of this control scheme,simulations are performed using the mathematical model of Esso Osaka tanker ship for set point regulation with and without disturbance and reference tracking.In addition,the simulation results are compared with that of a PID controller for further verification and validation.It has been shown that the proposed optimal controller can be used for the nonlinear ship steering with good rise time,zero steady state error and fast settling time.

Newton, Halley, Pell and the Optimal Iterative High-Order Rational Approximation of √<span style='margin-left:-2px;margin-right:2px;border-top:1px solid black'>N</span>

In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense of Pell’s equation p2 - Nq2 = k for some integer k, converging either alternatingly or oppositely.

履带车辆主动悬挂多点布置优化

为实现履带车辆主动悬挂减振性能和能耗达到综合最优,基于正交试验方法开展履带车辆主动悬挂多点布置优化设计。首先,建立了履带车辆悬挂系统动力学模型,并通过道路模拟试验验证了该模型的合理性;其次,开展了履带车辆悬挂系统正交试验,分析了4种典型路面下各子悬挂对悬挂系统减振性能影响的敏感性;最后,设计了主动悬挂作动器的6个布置方案,通过建立基于线性二次最优(linear quadratic regulator,简称LQR)控制的履带车辆主动悬挂动力学模型,分析了典型路面下各布置方案对悬挂系统减振性能的影响规律及能耗变化规律。结果表明,通过对履带车辆主动悬挂作动器的布置优化,可以实现悬挂减振性能和能耗之间的平衡。

STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

Discrete-time inverse linear quadratic optimal control over fnite time-horizon under noisy output measurements

In this paper,the problem of inverse quadratic optimal control over fnite time-horizon for discrete-time linear systems is considered.Our goal is to recover the corresponding quadratic objective function using noisy observations.First,the identifability of the model structure for the inverse optimal control problem is analyzed under relative degree assumption and we show the model structure is strictly globally identifable.Next,we study the inverse optimal control problem whose initial state distribution and the observation noise distribution are unknown,yet the exact observations on the initial states are available.We formulate the problem as a risk minimization problem and approximate the problem using empirical average.It is further shown that the solution to the approximated problem is statistically consistent under the assumption of relative degrees.We then study the case where the exact observations on the initial states are not available,yet the observation noises are known to be white Gaussian distributed and the distribution of the initial state is also Gaussian(with unknown mean and covariance).EM-algorithm is used to estimate the parameters in the objective function.The efectiveness of our results are demonstrated by numerical examples.

Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional.The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.Closedloop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.In this paper,the existence of an optimal closed-loop strategy for the system(also called the closedloop solvability of the problem)is characterized by the existence of a regular solution to the coupled two(generalized)Riccati equations,together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications

We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional.The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration.Semi closed-loop strategies are introduced,and following the dynamic programming approach in(Pham and Wei,Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics,2016),we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations.We present several financial applications with explicit solutions,and revisit,in particular,optimal tracking problems with price impact,and the conditional mean-variance portfolio selection in an incomplete market model.

Linear quadratic optimal controller for cable-driven parallel robots

In recent years, various cable-driven parallel robots have been investigated for their advantages, such as low structural weight, high acceleration, and large work- space, over serial and conventional parallel systems. However, the use of cables lowers the stiffness of these robots, which in turn may decrease motion accuracy. A linear quadratic （LQ） optimal controller can provide all the states of a system for the feedback, such as position and velocity. Thus, the application of such an optimal controller in cable-driven parallel robots can result in more efficient and accurate motion compared to the performance of classical controllers such as the proportional-integral-derivative controller. This paper presents an approach to apply the LQ optimal controller on cabledriven parallel robots. To employ the optimal control theory, the static and dynamic modeling of a 3-DOF planar cable-driven parallel robot （Feriba-3） is developed. The synthesis of the LQ optimal control is described, and the significant experimental results are presented and discussed.

Multistage Uncertain Random Linear Quadratic Optimal Control

In this paper,linear quadratic(LQ)optimal control problems are investigated for two types of uncertain random systems which consider the coefficient of the perturbed term as a constant vector or a vector-valued function of state vector and control vector.First,the uncertain random optimal control model is established under expected value criterion.Second,based on Bellman’s principle,recurrence equations are presented for settling such problem.Then by applying the recurrence equations and chance theory,the analytical expressions of the optimal results for the LQ problems are derived.Furthermore,some examples and an application are given to show the effectiveness of our results.

Identifiability and Solvability in Inverse Linear Quadratic Optimal Control Problems

In this paper, the inverse linear quadratic(LQ) problem over finite time-horizon is studied.Given the output observations of a dynamic process, the goal is to recover the corresponding LQ cost function. Firstly, by considering the inverse problem as an identification problem, its model structure is shown to be strictly globally identifiable under the assumption of system invertibility. Next, in the noiseless case a necessary and sufficient condition is proposed for the solvability of a positive semidefinite weighting matrix and its unique solution is obtained with two proposed algorithms under the condition of persistent excitation. Furthermore, a residual optimization problem is also formulated to solve a best-fit approximate cost function from sub-optimal observations. Finally, numerical simulations are used to demonstrate the effectiveness of the proposed methods.

半挂汽车列车挂车转向PSO-LQR控制器设计

针对低速转向时挂车跟踪牵引车轨迹性能较差的问题,设计了一种基于粒子群优化(particle swarm optimization, PSO)的挂车主动转向LQR控制器,探讨了不同权重矩阵获取方式对挂车转向控制效果的影响。验证了构建的挂车转向半挂汽车列车运动学模型的可靠性;设计了挂车的低速轨迹跟踪LQR控制器,利用PSO算法优化了LQR控制器的权重矩阵;研究了不同权重矩阵获取方式下的控制器性能。研究结果表明:经PSO算法优化后的LQR控制器能使挂车更快地进入稳定跟踪状态;当权重矩阵R分别取作0.1和1时,相比于由人为整定得到的权重矩阵Q对应的挂车跟踪误差,全局最优权重矩阵对应的挂车跟踪误差在单U形路径下分别减小26.1%和19.4%,在匝道螺旋路径下分别减小了40.9%和43.4%。