Separating two species of microalgae in photobioreactors in minimal time

This paper proposes a strategy to separate two strains of microalgae in minimal time. The control is the dilution rate of the continuous photobioreactor. The microalgae dynamics is described by the Droop’s model, taking into account the internal quota storage of the cells. Using Pontryagin’s principle, we develop a dilution-based control strategy that leads to the most efficient species separation in minimal time. A numerical optimal synthesis –based on direct optimization methods– is performed throughout the paper, in order to determine the structure of the optimal feedback-control law, which is bang-singular. Our numerical study reveals that singular arcs play a key role in the optimization problem since they allow the optimal solution to be close to an associated static optimal control problem. A resulting turnpike-like behavior, which characterizes the optimal solution, is highlighted throughout this work.     

Fixed-structure controller design for systems with actuator amplitude and rate non-linearities

In this paper we develop output feedback controllers and fixed-order (i.e., full- and reduced-order) dynamic compensators for systems with actuator amplitude and rate saturation constraints. The proposed design methodology employs a rate limiter as part of the controller architecture. The problem of simultaneous control amplitude and rate saturation is embedded within an optimization problem by constructing a Riccati equation whose solution guarantees closed-loop global/local asymptotic stability in the face of sector bounded actuator amplitude and rate non-linearities. Application of the proposed framework is demonstrated via a flight control example involving actuator amplitude and rate saturation constraints.     

Inf–sup control of discontinuous piecewise affine systems

This paper considers the worst‐case optimal control of discontinuous piecewise affine (PWA) systems, which are subjected to constraints and disturbances. We seek to pre‐compute, via dynamic programming, an explicit control law for these systems when a PWA cost function is utilized. One difficulty with this problem class is that, even for initial states for which the value function of the optimal control problem is finite, there might not exist a control law that attains the infimum. Hence, we propose a method that is guaranteed to obtain a sub‐optimal solution, and where the degree of sub‐optimality can be specified a priori. This is achieved by approximating the underlying sub‐problems with a parametric piecewise linear program. Copyright © 2008 John Wiley & Sons, Ltd.     

Real‐time model predictive control based on dual gradient projection: Theory and fixed‐point FPGA implementation

This paper proposes a method to design robust model predictive control (MPC) laws for discrete‐time linear systems with hard mixed constraints on states and inputs, in case of only an inexact solution of the associated quadratic program is available, because of real‐time requirements. By using a recently proposed dual gradient‐projection algorithm, it is proved that the discrepancy of the optimal control law as compared with the obtained one is bounded even if the solver is implemented in fixed‐point arithmetic. By defining an alternative MPC problem with tightened constraints, a feasible solution is obtained for the original MPC problem, which guarantees recursive feasibility and asymptotic stability of the closed‐loop system with respect to a set including the origin, also considering the presence of external disturbances. The proposed MPC law is implemented on a field‐programmable gate array in order to show the practical applicability of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal Control of Underactuated Nonholonomic Mechanical Systems

In this paper, we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, underactuated mechanical systems. In particular, we aim to minimize the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and underactuation. Nonholonomic constraints restrict the evolution of the system to a distribution on the manifold. The nonholonomic connection is used to express the constrained equations of motion. Many robotic systems are underactuated since control inputs are usually applied through the robot's internal configuration space only. While we do not consider symmetries with respect to group actions in this paper, the fact that the system is underactuated is taken into account in our problem formulation. This allows one to compute reaction forces due to any inputs applied in directions orthogonal to the constraint distribution. We illustrate our ideas by considering a simple example on a three-dimensional manifold, including obstacle avoidance using the method of navigation functions.     

A sub-optimal algorithm to synthesize control laws for a network of dynamic agents

We study the synthesis problem of a linear quadratic regulator (LQR) controller when the matrix describing the control law is constrained to lie in a particular vector space. Our motivation is the use of such control laws to stabilize networks of autonomous agents in decentralized fashion, with the information flow being dictated by the constraints of a pre-specified topology. In this paper, we consider the finite-horizon version of the problem and provide both a computationally intensive optimal solution and a sub-optimal solution that is computationally more tractable. Then we apply the technique to the decentralized vehicle formation control problem. It is numerically illustrated that while the loss in performance due to the use of the sub-optimal solution is not huge, the topology can have a large effect on performance.     

Robust attitude tracking control for a rigid spacecraft under input delays and actuator errors

Attitude control of a rigid spacecraft under input delays, disturbances, parameter uncertainties, actuator errors, and constraints is a challenging problem. In this paper, these problems are considered simultaneously, and a robust control approach to attitude tracking of a rigid spacecraft is exploited. The design methodology is based on three steps: (1) compensating input delays by using the backstepping technique, (2) design of a disturbance observer for the delayed system by using the super-twisting algorithm to estimate unknown internal and external disturbances, then adding a feedforward compensation law based on the estimated signal to the backstepping controller to attenuate the effects of disturbances, (3) employing a robust least-square scheme to map the specified control command on the redundant actuators in the presence of actuator error, including actuator magnitude deviation and misalignment, with regard to actuator amplitude and rate constraints. The effectiveness of the proposed algorithm is shown by various numerical simulations.     

Decentralized tube‐based model predictive control of uncertain nonlinear multiagent systems

This paper addresses the problem of decentralized tube‐based nonlinear model predictive control (NMPC) for a general class of uncertain nonlinear continuous‐time multiagent systems with additive and bounded disturbance. In particular, the problem of robust navigation of a multiagent system to predefined states of the workspace while using only local information is addressed under certain distance and control input constraints. We propose a decentralized feedback control protocol that consists of two terms: a nominal control input, which is computed online and is the outcome of a decentralized finite horizon optimal control problem that each agent solves at every sampling time, for its nominal system dynamics; and an additive state‐feedback law which is computed offline and guarantees that the real trajectories of each agent will belong to a hypertube centered along the nominal trajectory, for all times. The volume of the hypertube depends on the upper bound of the disturbances as well as the bounds of the derivatives of the dynamics. In addition, by introducing certain distance constraints, the proposed scheme guarantees that the initially connected agents remain connected for all times. Under standard assumptions that arise in nominal NMPC schemes, controllability assumptions, communication capabilities between the agents, it is guaranteed that the multiagent system is input‐to‐state stable with respect to the disturbances, for all initial conditions satisfying the state constraints. Simulation results verify the correctness of the proposed framework.     

The cost optimal solution of the multi-constrained multicast routing problem

In this paper, we define the cost optimal solution of the multi-constrained multicast routing problem. This problem consists in finding a multicast structure that spans a source node and a set of destinations with respect to a set of constraints, while minimizing a cost function. This optimization is particularly interesting for multicast network communications that require Quality of Service (QoS) guarantees. Finding such a structure that satisfies the set of constraints is an NP-hard problem. To solve the addressed routing problem, most of the proposed algorithms focus on multicast trees. In some cases, the optimal spanning structure (i.e. the optimal multicast route) is neither a tree nor a set of trees nor a set of optimal QoS paths. The main result of our study is the exact identification of this optimal solution. We demonstrate that the optimal connected partial spanning structure that solves the multi-constrained multicast routing problem always corresponds to a hierarchy, a recently proposed generalization of the tree concept. We define the directed partial minimum spanning hierarchies as optimal solutions for the multi-constrained multicast routing problem and analyze their relevant properties. To our knowledge, our paper is the first study that exactly describes the cost optimal solution of this NP-hard problem.     

On constrained infinite-time linear quadratic optimal control

This work presents a solution to the infinite-time linear quadratic optimal control (ITLQOC) problem with state and control constraints. It is shown that a single, finite dimensional, convex program of known size can yield this solution. Properties of the resulting value function, with respect to initial conditions, are also established and are shoen to be useful in determining the aforementioned problem size. An example illustrating the method is finally presented.     

Optimal feedback control scheme helping managers to adjust aggregate industrial resources

This paper deals with the problem of stochastic production planning with chance constraints on the decision variables. A solution to such problems serves as an invaluable tool for managers in the process of developing a production plan to their companies. Because finding the true optimal solution is not a trivial task, one usually resorts to approximate solutions. One common approach is to transform the original problem into the equivalent mean problem. The mean solution, however, is essentially deterministic and is not responsive to the actual behavior of the fluctuating demand. Then, to improve the mean solution, a feedback control law is proposed. The control law is analyzed and a method to compute the feedback gain is discussed. An example is presented to illustrate the main results.     

Conditions under which suboptimal nonlinear MPC is inherently robust

We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.     

Disturbance tolerance and H ∞ control of port-controlled hamiltonian systems in the presence of actuator saturation

This paper investigates the disturbance tolerance and H _∞ control of multi-input Port-Controlled Hamiltonian (PCH) systems in the presence of actuator saturation which may be not openloop stable. A simple condition is derived under which trajectories starting from the origin will remain inside an ellipsoid. The disturbance tolerance ability of the closed-loop system under a given feedback control law is measured by the size of this ellipsoid. Based on the above mentioned condition, the problem of disturbance tolerance can be expressed in the form of the linear matrix inequalities (LMIs) optimization problem with constraints. In addition, an H _∞ control approach is presented to attenuate the disturbances, and disturbance rejection ability in terms of L ₂ gain is also determined by the solution of an LMI optimization problem. Study of an illustrative example with simulations shows the effectiveness of the methods proposed.     

On an inverse problem in optimal control

The problem of optimizing linear-stationary control systems subject to quadratic performance indices and bounded control effort constraints is studied. This problem was first proposed by Letov [1]. His solution, however, has been found to be valid only in some special cases [3]. In this paper, necessary and sufficient conditions for the existence of the optimal control laws of the saturation type are developed. It is shown that the weighting factors in the performance index have to be chosen in a particular manner, and the plants have to satisfy certain conditions to yield an optimal saturation type control law. As the weighting on the control effort in the performance index is decreased, the optimal solution approaches a bang-bang type singular control.     

Resolving actuator redundancy—optimal control vs. control allocation

This paper considers actuator redundancy management for a class of overactuated nonlinear systems. Two tools for distributing the control effort among a redundant set of actuators are optimal control design and control allocation. In this paper, we investigate the relationship between these two design tools when the performance indexes are quadratic in the control input. We show that for a particular class of nonlinear systems, they give exactly the same design freedom in distributing the control effort among the actuators. Linear quadratic optimal control is contained as a special case. A benefit of using a separate control allocator is that actuator constraints can be considered, which is illustrated with a flight control example.     

An algorithm for simultaneous stabilization using decentralizedconstant gain output feedback

An algorithm which solves, numerically, the simultaneous stabilization problem using a constant gain decentralized control law is presented. The algorithm is determined from the necessary conditions for minimizing an optimal control problem. The optimal control problem consists of n plant models each with its own cost function. The costs are summed to create an average cost function and equality constraints are added to yield the decentralized control structure. The algorithm can start with any stabilizing full state feedback gain for each model and will converge to the optimal constant feedback gain for all models assuming a solution exists. Examples of the algorithm are given for Kharitonov synthesis and optimal gain scheduled control law synthesis using output feedback     

计算机数控系统光滑时间最优轨迹规划

基于控制向量参数化(CVP)方法, 研究了计算机数控(CNC)系统光滑时间最优轨迹规划方法. 通过在规划问题中引入加加速度约束, 实现轨迹的光滑给进. 引入时间归一化因子, 将加加速度约束的时间最优轨迹规划问题转化为固定时间的一般性最优控制问题. 以路径参数对时间的三阶导数(伪加加速度)和终端时刻为优化变量, 并采用分段常数近似伪加加速度, 将最优控制问题转化为一般的非线性规划(NLP)问题进行求解. 针对加加速度、加速度等过程不等式约束, 引入约束凝聚函数, 将过程约束转化为终端时刻约束, 从而显著减少约束计算. 构造目标和约束函数的Hamiltonian函数, 利用伴随方法获得求解NLP问题所需的梯度.     

A problem in jointly optimal production and advertising decisions

In this paper, a production control model proposed by Holt, Modigliani, Muth and Simon and a marketing model suggested by Nerlove and Arrow, are combined to produce an optimal control problem with two state variables, two control variables and inequality constraints on these variables. The optimal production rate and advertising expenditures over time are determined using the maximum principle. Due to the complex nature of the necessary conditions, the numerical technique known as the initial-value shooting method, is used to generate optimal trajectories. A numerical example is presented to illustrate the solution.     

Optimal control for zinc solution purification based on interacting CSTR models

In zinc solution purification process, zinc dust is used to remove impurity ions in several serial stirred tank reactors. It is modelled by using several interacting continuously stirred tank reactor (ICSTR) systems with multiple time delays. Some unknown parameters of the ICSTR model are identified based on experimental data collected from a zinc production factory in China. Then a time delayed optimal control problem with continuous inequality constraints is constructed for the zinc solution purification. A numerical computational algorithm is developed, based on the control parameterization method, to design an optimal control law to ensure that a sufficiently low level of effluent impurities is achieved with the least zinc dust consumption. Finally, numerical simulation is carried out. The results obtained indicate that the effect of the optimal control for zinc solution purification is highly satisfactory.