An algorithm for a solution of a stochastic adaptive linear quadratic optimal control problem

A variational approach is taken to derive optimality conditions for a discrete-time linear quadratic adaptive stochastic optimal control problem. These conditions lead to an algorithm for computing optimal control laws which differs from the dynamic programming algorithm.     

Two-level stochastic LQ control problem for a dynamic system

Two-level, hierarchical, partially decentralized control for a large-scale system is considered. It is seen that for an LQ problem the optimal control laws are linear functions of the state estimates. The filter equations for some state estimates, determined by a coordinator and local controllers co-operating in a two-level control structure, are derived, A discussion on the usefulness of the application of particular local controllers is given, as well as a numerical example.     

离散双输出反馈控制的最优协调性合成设计

本文以离散二次型最优控制理论为基础，研究了线性时不变系统在双输出反馈控制情况下控制作用的协调性合成设计问题，构造了一个数值迭代算法来求解相应的最侉 控制规律。通过产例计算和对比分析，既证明了该方法的收敛性和有效性又说明了这种控制问题的一些本质特征。     

Parameter-sensitivity study for a linear-quadratic control problem with random state coefficients

Optimal feedback-control laws generally cannot be obtained in closed form for stochastic control problems. The characterizations which have been obtained for several simplified problems have provided valuable insight into the properties of optimal feedback-control laws as well as providing guidance for the construction of suboptimal control laws. In this paper an analytical and computer simulated parameter-sensitivity study is presented for the optimal feedback-control law and dynamic-programming optimality equations associated with a discrete-time, finite-horizon, linear-quadratic control problem with random state coefficients. One interesting characteristic revealed by analysis is the existence of simple linear relationships between parameter sensitivities for the optimal feedback-control selections and for the corresponding cost-to-go expressions.     

A simple dual control problem with an analytical solution

A stochastic control problem for which the optimal dual control law can be calculated analytically is given. The system is a four state Markov chain with transition probabilities that depend on the control variable. The performance of the optimal dual control law and of some suboptimal control laws are calculated and compared.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

Design of nonlinear regulators for linear plants

A procedure for designing stable nonlinear control laws for linear plants is presented. The design is based on an inverse optimal control problem with the closed form controller being directly given by the solution of a set of linear algebraic equations. The nonlinear controllers can be used to produce saturation effects on particular states, and an example is included to illustrate the results.     

Dynamic control of coordinated redundant robots with torque optimization

The problem of controlling a system of coordinated redundant robots with torque optimization based on joint redundancy is addressed. Local and global optimal control laws, both minimizing joint torque loading, are developed. A general method of load distribution among the coordinated robots is also proposed. The control problem is to regulate the motion of the object held by the coordinated robots and the internal force generated as a result of constraints on the object. The errors in the object motion and internal force converge asymptotically to zero under the proposed optimal control laws, when exact knowledge of the dynamic models is assumed. Furthermore, the robustness of the proposed method to model uncertainty is also analyzed. The motion and internal force errors are uniformly ultimately bounded under the proposed optimal controllers, when uncertainty in the dynamic models is assumed to exist.     

基于T-S模型的倒立摆最优保性能模糊控制

对一类具有范数有界参数不确定性T-S模糊模型系统,采用状态反馈的并行分布补偿器(PDC)结构,基于线性矩阵不等式处理方法,研究了其最优保性能模糊控制律的设计问题.导出了保性能模糊控制律存在的条件,通过求解一个凸优化问题给出了最优保性能模糊控制律的设计方法,并用此方法设计了倒立摆系统的最优保性能模糊控制器.仿真实验验证了该设计方法的有效性.     

Some new results on optimal and suboptimal regulators of the LQproblem with output feedback

The LQ (linear quadratic) regulator problem with output feedback is considered. In the problem studies an output feedback control is determined such that the cost is J*, the optimal cost for the LQ problem with state feedback, or βj* with some constant β>1. A set of sufficient conditions is given for determining such optimal and suboptimal output feedback control laws     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

Optimal control laws for lot-sizing and timing of jobs on a single production facility

The optimal control of a single machine processing a certain number of jobs and modeled as a discrete-event dynamic system is considered. The number of jobs and their sequence are fixed, whereas their timing and sizes represent the control variables of the system. The objective function to be optimized is a weighted sum of the quadratic earliness and tardiness of each job, and of the quadratic deviations of job lot sizes and actual machine service speeds from those specified by the production demand and by the regular machine speeds. An optimization problem with quadratic cost function and nonlinear constraints is stated and formalized as a multistage optimal control problem. Necessary conditions to be satisfied by an optimal control sequence are derived. A simpler model is also considered in which the machine speeds are fixed; in this case, the control problem is solved by a procedure making use of dynamic programming techniques. The optimal control laws at each stage are thus obtained.     

Optimal control of stochastic systems with interrupted observations

An optimal control problem is considered for a nonlinear stochastic system with an interrupted observation mechanism that is characterized in terms of a jump Markov process taking on the values 0 or 1. The state of the system is described by a diffusion process, but the observation has components modulated by the jump process. The admissible control laws are smooth functions of the observation. Using the calculus of variations, necessary conditions on optimal controls are derived. These conditions amount to solving a set of four coupled nonlinear partial differential equations. A numerical procedure for solving these equations is suggested and an example dealt with numerically.     

Nonlinear feedback solution to a bounded Brachistochrone problem in a reduced state space

The optimal nonlinear feedback law for both an unbounded and bounded Brachistochrone problem is given. These control laws describe the Brachistochrone problem in a very simple manner. These laws are derived by using a dimensionless set of state variables which spans a lower dimensional space than the original state space. Also using this reduced state space a two-dimensional instead of three-dimensional field of extremals is constructed for a space bounded Brachistochrone problem. This illustrates that the state space may be reduced so that the storage of a field of extremal trajectories will not exceed the storage capacity of a practical on-board computer. In this way, optimal feedback control may become a practical technique for a larger class of nonlinear systems.     

Static team problems--Part II: Affine control laws, projections, algorithms, and the LEGT problem

Team problems over the set of affine team control laws are solved as constrained parameter optimization problems. This methodology eases both the notational and computational difficulties. For the LQGT (linear-quadratic-Gaussian team) and the LEGT (linear-exponential-Gaussian team) problems, the constrained parameter optimization approach leads to the representation of the optimal team control gains as explicit projections of the optimal centralized gains. The projection representation offers insight into the team solution and suggests an algorithm for computing the optimal gains for the LEGT problem. The LEGT problem behavior is compared to LQGT behavior for an example which consists of a static team with garbled decision implementation.     

Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin’s minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax–Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin’s minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions.     

基于动态规划的约束优化问题多参数规划求解方法及应用

结合动态规划和单步多参数二次规划, 提出一种新的约束优化控制问题多参数规划求解方法. 一方面能得到约束线性二次优化控制问题最优控制序列与状态之间的显式函数关系, 减少多参数规划问题求解的工作量; 另一方面能够同时求解得到状态反馈最优控制律. 应用本文提出的多参数二次规划求解方法, 建立无限时间约束优化问题状态反馈显式最优控制律. 针对电梯机械系统振动控制模型做了数值仿真计算.     

Bi-Objective Optimal Control Modification Adaptive Control for Systems with Input Uncertainty

This paper presents a new model-reference adaptive control method based on a bi-objective optimal control formulation for systems with input uncertainty. A parallel predictor model is constructed to relate the predictor error to the estimation error of the control effectiveness matrix. In this work, we develop an optimal control modification adaptive control approach that seeks to minimize a bi-objective linear quadratic cost function of both the tracking error norm and the predictor error norm simultaneously. The resulting adaptive laws for the parametric uncertainty and control effectiveness uncertainty are dependent on both the tracking error and the predictor error, while the adaptive laws for the feedback gain and command feedforward gain are only dependent on the tracking error. The optimal control modification term provides robustness to the adaptive laws naturally from the optimal control framework. Simulations demonstrate the effectiveness of the proposed adaptive control approach.      

Finite‐Time Optimal Formation Control for Second‐Order Multiagent Systems

This paper studies the problem of finite‐time optimal formation control for second‐order multiagent systems in situations where the formation time and/or the cost function need to be considered. The finite‐time optimal formation control laws are proposed for the cases with or without a leader, respectively. For the case of control being constrained, the time optimal formation problem is considered and an algorithm is designed to derive a feasible solution for the problem concerned. Although the feasible solution may not be optimal, it can provide a lower bound for time for the formation problem with control constraints. Once the given formation time is lower than this bound, the control constraints cannot be ensured. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.