Differential flatness based nonlinear predictive control of fed-batch bioreactors

A trajectory optimization scheme based on the property of differential flatness is proposed in this paper. A dynamic optimization problem is transformed into a lower dimensional nonlinear programming problem through the use of flat outputs. This optimization approach is demonstrated in the repeated optimization of nonlinear dynamic systems under feedback in an approach similar to nonlinear model predictive control. This approach is illustrated on two examples involving biomass optimization and product optimization. Optimization under feedback is studied for the nominal problem and the case where uncertainty is present. The proposed scheme is also used in conjunction with a nonlinear Luenberger observer to generate the optimal trajectories under parametric uncertainty.     

Active probing for information in control systems with quantizedstate measurements: a minimum entropy approach

In this paper the effect of state quantization in scaler discrete-time linear control systems is studied by analyzing the system as a partially observed stochastic system. The problem of optimal state information gathering and filtering is investigated using information theoretic measures and formulating the state estimation problem as an entropy optimization problem. The active probing effect of the feedback control is thoroughly studied. Optimal feedback controls which minimize various types of entropy costs are determined, and it is shown that this problem is equivalent to an optimal control problem for a controlled Markov chain     

Depth control of autonomous underwater vehicles using indirect robust control method

In this article, we propose a robust depth control design scheme for autonomous underwater vehicles (AUVs) in the presence of hydrodynamic parameter uncertainties and disturbances. The controller is designed via a new indirect robust control method that handles the uncertainties by formulating the uncertainty bounds into the cost functional and then transforming the robust control problem into an equivalent optimal control problem. Both robust asymptotic stability and optimality can be achieved and proved with this new formulation. The θ-D method is utilised to solve the resultant nonlinear optimal control problem such that an approximate closed-form feedback controller can be obtained and thus is easy to implement onboard without intensive computation load. Simulation results demonstrate that robust depth control is accomplished under the system parameter uncertainties and disturbances with small control fin deflection requirement.     

Practical optimal state feedback control law for continuous-time switched affine systems with cyclic steady state

In this article, a method for computing an optimal state feedback control law for continuous-time switched affine systems exhibiting cyclic behaviour in steady state is presented. The hybrid solutions are deduced from the Fillipov solutions. It is shown that the optimal trajectory synthesis implies to determine singular arcs. Algebraic conditions are given to obtain these particular arcs of the trajectory. A numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two-point boundary value problem occurring in optimal control synthesis. The interpolation of the solutions set, through a neural network, yields a state feedback control law. Several examples in the power converters field show the feasibility and the efficiency of the method.     

Optimal control of switching systems

This paper considers an optimal control problem for a switching system. For solving this problem we do not make any assumptions about the number of switches nor about the mode sequence, they are determined by the solution of the problem. The switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter. It is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system. The relationship between the two sets of trajectories (1) motivates the shift of focus from the original problem to the more general one and (2) underlies the engineering relevance of the study of the second problem. Sufficient and necessary conditions for optimality are formulated for the second optimization problem. If they exist, bang-bang-type solutions of the embedded optimal control problem are solutions of the original problem. Otherwise, suboptimal solutions are obtained via the Chattering Lemma.     

Time Optimal Swing-Up of the Planar Pendulum

This paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart. This mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open-loop controls. Relying on the theory of optimal synthesis, we provide a discontinuous feedback giving optimal solutions for any initial data. The approach is that of geometric control theory.     

Dynamic supply chain scheduling

Based on a combination of fundamental results of modern optimal program control theory and operations research, an original approach to supply chain scheduling is developed in order to answer the challenges of dynamics, uncertainty, and adaptivity. Both supply chain schedule generation and execution control are represented as an optimal program control problem in combination with mathematical programming and interpreted as a dynamic process of operations control within an adaptive framework. Hence, the problems and models of planning, scheduling, and adaptation can be consistently integrated on a unified mathematical axiomatic of modern control theory. In addition, operations control and flow control models are integrated and applicable for both discrete and continuous processes. The application of optimal control for supply chain scheduling becomes possible by formulating the scheduling model as a linear non-stationary finite-dimensional controlled differential system with the convex area of admissible control and a reconfigurable structure. For this model class, theorems of optimal control existence can be used regarding supply chain scheduling. The essential structural property of this model are the linear right parts of differential equations. This allows applying methods of discrete optimization for optimal control calculation. The calculation procedure is based on applying Pontryagin’s maximum principle and the resulting essential reduction of problem dimensionality that is under solution at each instant of time. The gained insights contribute to supply chain scheduling theory, providing advanced insights into dynamics of the whole supply chains (and not any dyadic relations in them) and transition from a partial “one-way” schedule optimization to the feedback loop-based dynamic and adaptive supply chain planning and scheduling.     

Optimal trajectories in brachistochrone problem with Coulomb friction

A two-parameter family of optimal curves in the brachistochrone problem in the case of Coulomb friction is found. The problem is represented in the form of the standard time minimization control problem. The normal component of the support reaction is used as control. It turned out that the formula for the optimal control, which does not include adjoint variables, has a singularity at the zero motion velocity. A system of ordinary differential equations is derived for which the solution of the Cauchy initial value problem makes it possible to obtain optimal trajectories that have a vertical tangent at the initial point. The self-similarity property of such trajectories is proved. It is shown how this property can be used to obtain by scaling all optimal trajectories from the set of optimal trajectories with fixed initial conditions and different terminal slope angles of the tangent.     

A robust model predictive control algorithm for incrementally conic uncertain/nonlinear systems

This paper presents a robustly stabilizing model predictive control algorithm for systems with incrementally conic uncertain/nonlinear terms and bounded disturbances. The resulting control input consists of feedforward and feedback components. The feedforward control generates a nominal trajectory from online solution of a finite‐horizon constrained optimal control problem for a nominal system model. The feedback control policy is designed off‐line by utilizing a model of the uncertainty/nonlinearity and establishes invariant ‘state tubes’ around the nominal system trajectories. The entire controller is shown to be robustly stabilizing with a region of attraction composed of the initial states for which the finite‐horizon constrained optimal control problem is feasible for the nominal system. Synthesis of the feedback control policy involves solution of linear matrix inequalities. An illustrative numerical example is provided to demonstrate the control design and the resulting closed‐loop system performance. Copyright © 2010 John Wiley & Sons, Ltd.     

基于控制参数化的注塑工业过程最优反馈控制方法

本文针对一类典型的注塑工业过程系统, 研究了注塑填充过程中产生的熔体流动速度最优跟踪控制问题, 提出了一种基于控制参数化的计算最优反馈控制器设计方法以实现注塑过程中熔融聚合物流动前沿位移的最优跟 踪控制, 进而达到改善注塑零件性能的高效生产目标. 首先, 面向注塑工艺复杂生产过程建立了动态过程系统数学 模型, 提出了注塑机内部熔融聚合物流动前沿位置的动态最优跟踪控制问题; 其次, 设计了一种多级反馈控制律, 通 过控制参数化方法将控制反馈核进行了参数化表示, 将控制器设计问题转化为一序列最优参数决策问题; 然后, 通 过状态灵敏度方程分析方法, 求解出了目标函数及约束条件关于决策变量参数梯度信息的显式表达式, 并基于所提 供的梯度信息结合序列二次规划算法进行了高效优化迭代求解; 最后, 通过实验仿真验证了本文所提出的最优反 馈控制器设计方法的可行性和有效性.     

不确定离散系统的最优保性能控制*

对一类具有范数有界时变参数不确定性的离散时间线性系统和一个二次型性能指标,研究其最优保性能状态反馈控制律的设计问题。     

Incorporating uncertainty in optimal decision making: Integrating mixed integer programming and simulation to solve combinatorial problems

We introduce a novel methodology that integrates optimization and simulation techniques to obtain estimated global optimal solutions to combinatorial problems with uncertainty such as those of facility location, facility layout, and scheduling. We develop a generalized mixed integer programming (MIP) formulation that allows iterative interaction with a simulation model by taking into account the impact of uncertainty on the objective function value of previous solutions. Our approach is generalized, efficient, incorporates the impact of uncertainty of system parameters on performance and can easily be incorporated into a variety of applications. For illustration, we apply this new solution methodology to the NP-hard multi-period multi-product facility location problem (MPP-FLP). Our results show that, for this problem, our iterative procedure yields up to 9.4% improvement in facility location-related costs over deterministic optimization and that these cost savings increase as the variability in demand and supply uncertainty are increased.     

Open-loop implementation of feedbacks in parametric optimization problems for incompletely determined systems with distributed parameters

A constructive method for the design and real-time implementation of parameterized open-loop controls in the optimal feedback control of ensembles of trajectories of incompletely determined parabolic systems with distributed parameters is proposed. This method uses reduction to semiinfinite optimization problems, alternance properties of their solutions, and additional information about fundamental laws of the application domain. The practical application of the results is illustrated by the example of controlling nonstationary temperature fields under multiple disturbances, which is a problem of independent interest.     

Robust model predictive control using tubes

A form of feedback model predictive control (MPC) that overcomes disadvantages of conventional MPC but which has manageable computational complexity is presented. The optimal control problem, solved on-line, yields a ‘tube’ and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty; computational complexity is linear (rather than exponential) in horizon length. Asymptotic stability of the controlled system is established.     

New Formulation of Dynamic Output Feedback Robust Model Predictive Control with Guaranteed Quadratic Boundedness

The dynamic output feedback robust model predictive controller for a system with both polytopic uncertainty and bounded disturbance is addressed in this paper. This controller utilizes a main optimization problem to find the control law and a simple auxiliary optimization problem to refresh the bounds on the true state. The main optimization problem, which is not necessarily solved at each sampling instant, achieves the near‐optimal solution. The auxiliary optimization, which is solved at each sampling instant, is followed with a simple criterion which determines whether or not to solve the main optimization problem at the next sampling time. By applying the proposed method, the augmented state of the closed‐loop system is guaranteed to converge to the neighborhood of the equilibrium point.     

Stabilizing a linear system with Saturation Through optimal control

We construct a continuous feedback for a saturated system x(t)=Ax(t)+B/spl sigma/(u(t)). The feedback renders the system asymptotically stable on the whole set of states that can be driven to 0 with an open-loop control. The trajectories of the resulting closed-loop system are optimal for an auxiliary optimal control problem with a convex cost and linear dynamics. The value function for the auxiliary problem, which we show to be differentiable, serves as a Lyapunov function for the saturated system. Relating the saturated system, which is nonlinear, to an optimal control problem with linear dynamics is possible thanks to the monotone structure of saturation.     

Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems

The problem of optimal control of automatic deterministic discrete systems in conditions of parametric uncertainty is considered. The state change (switchover) of a system is described by a recurrent equation in which instant multiple switchovers are admitted. The times and the number of switchovers are not specified in advance—they are found as a result of optimization of a functional. The initial state of the system is not known exactly, but the set of possible initial states is known; therefore, problems of average optimal control and guaranteed optimal control of bunches of trajectories are formulated. For solving these problems, it is suggested to use the principle of separation: namely, to control the bunch of trajectories by applying a control optimal for one specially chosen trajectory of the bunch. The resulting control of the bunch is suboptimal, but it may be satisfactory in practice. On the basis of sufficient conditions for optimality of a full-feedback control, an algorithm for constructing a suboptimal control in conditions of uncertainty is developed. Counterexamples of linear–quadratic problems of control of automaton-type systems in which the principle of separation fails are presented.     

可变服务率模式下基于需求驱动的传送带给料加工站系统的优化控制

本文主要研究可变服务率模式下基于需求驱动的传送带给料加工站(CSPS)系统的优化控制问题,主要目标是对系统的随机优化控制问题进行建模和提供解决方案.论文以缓冲库和成品库剩余容量为联合状态,以站点前视距离和工件服务率为控制变量,将其最优控制问题描述为半马尔科夫决策过程(SMDP)模型.该模型为利用策略迭代等方法求解系统在平均准则或折扣准则下的最优控制策略提供了理论基础,特别地,据此可引入基于模拟退火思想的Q学习算法等优化方法来寻求近似解,以克服理论求解过程中的维数灾和建模难等困难.仿真结果说明了本文建立的数学模型及给出的优化方法的有效性.     

Run-to-run optimization via control of generalized constraints

Run-to-run optimization methodologies exploit the repetitive nature of batch processes to determine the optimal operating policy in the presence of uncertainty. In this paper, a parsimonious parameterization of the inputs is used and the decision variables of the parameterization are updated on a run-to-run basis using a feedback control scheme which tracks signals that are invariant under uncertainty. In this run-to-run framework, terminal constraints of the optimization problem and cost sensitivities constitute the invariant signals. The methodology is conceived to improve the cost function from batch-to-batch without constraint violation.     

Reducing energy consumption through trajectory optimization for a metro network

The techniques of optimal control and optimization are applied to a practical problem of reducing energy consumed by the Montreal Metro (subway) system. The problem considered is the determination of tunnel trajectories in the "equivalent" vertical plane when trains traveling in both directions must follow the same trajectory. The problem is first formulated as a control problem with control and state constraints. Then, under certain simplifying assumptions, an heuristic method employing a direct search algorithm is presented and used in the trajectory optimization. The trajectories are optimized to reduce the sum of the energy consumed by the trains traveling in both directions on the trajectory. The results show an average reduction of 7.73 percent in energy consumption as compared with existing trajectories. The trajectories found using the method presented here will be followed in future tunnel construction.