Optimal guaranteed control of linear systems under disturbances

This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 198–205, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-000771a.     

Optimal disturbance rejection control for nonlinear impulsive dynamical systems

In this paper, we develop an optimality-based framework for addressing the problem of nonlinear–nonquadratic hybrid control for disturbance rejection of nonlinear impulsive dynamical systems with bounded exogenous disturbances. Specifically, we transform a given nonlinear–nonquadratic hybrid performance criterion to account for system disturbances. As a consequence, the disturbance rejection problem is translated into an optimal hybrid control problem. Furthermore, the resulting optimal hybrid control law is shown to render the closed-loop nonlinear input–output map dissipative with respect to general supply rates. In addition, the Lyapunov function guaranteeing closed-loop stability is shown to be a solution to a steady-state hybrid Hamilton–Jacobi–Isaacs equation and thus guaranteeing optimality.     

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

Optimal Infinite-Horizon Multicriteria Feedback Control of Stationary Systems with Minimax Objectives and Bounded Disturbances

This article presents a methodology for exploring the solution surface in a class of multicriteria infinite-horizon closed-loop optimal control problems with bounded disturbances and minimax objectives. The maximum is taken with respect to both time and all sequences of disturbances; that is, the value of a criterion is the maximal stage cost for the worst possible sequence of disturbances. It is assumed that the system and the cost functions are stationary. The proposed solution method is based on reference point approach and inverse mapping from the space of objectives into the space of control policies and their domains in state space.     

Optimal sliding mode control of single degree-of-freedom hysteretic structural system

This paper addresses the controller design problem of a nonlinear single degree-of-freedom structural system excited by the earthquake. Bouc–Wen model, as an efficient hysteresis modeling method, is used to model the system nonlinearity. Sliding mode control (SMC), due to its robustness in dealing with uncertainty, is utilized as the main control strategy. An optimal sliding surface is presented which minimizes the displacement and control force in terms of a quadratic cost function. Two numerical examples are given to illustrate the effectiveness of the proposed strategy subject to three earthquakes of El-Centro, Rinaldi and Kobe. Simulation results show a significant and considerable reduction in structural response and indicate that the performance of suggested optimal SMC strategy is remarkable.     

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.     

Advanced Optimal Feedback Control for Tracking of Real-Life Applications

Calculating the open–loop solution of an optimal control problem is just the first step to cope with the practical realization of real life applications. Feedback controllers, like the classical Linear Quadratic Regulator (LQR), are needed to compensate pertubations appearing in reality. Although these controllers have proven to be a powerful tool in many applications and to be robust enough to countervail most differences between simulation and practice, they are not optimal if disturbances in the system data occur. If these controllers are applied in a real process, the possibility of data disturbances force recomputing the feedback control law in real–time to preserve stability and optimality, at least approximately. For this purpose, variations of the classical closed–loop controller with the extention to a trackingtype controller are analysed by means of an industrial application of container cranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Noise Rejection in Structural Analysis by Means of Generalized Sampled-Data Hold Functions

In this paper, a technique for optimal noise rejection, based on generalized sampled-data hold functions is applied to the control of civil engineering structures. The technique consists in suitably modulating the sampled outputs of the system under control by periodically varying functions in order to attenuate the effect of the disturbances on the system states to an acceptable level, by minimizing a quadratic cost function. This minimization is performed by feeding back the outputs of the system, which are assumed to be corrupted by measurement noise. Moreover, in the present paper, the robustness properties of the GSHF based optimal regulator is analyzed and guaranteed stability margins, expressed in terms of elementary cost and system matrices, are proposed for such a type of optimal regulators. The effectiveness of the method is demonstrated by various simulation results. The results of the paper can be used to assess the detrimental effect of noise on the closed-loop system and the tradeoff involved in assuring good sampled-data performance and sufficient robustness.     

Optimal feedback control of the forced van der Pol system

A simple feedback control strategy for chaotic systems is investigated using the forced van der Pol system as an example. The strategy regards chaos control as an optimization problem, where the maximum magnitude Floquet multiplier of a target unstable periodic orbit (UPO) is used as a cost function that needs to be minimized. Thus, the method obtains the optimal control gain in terms of the stability of the target UPO. This strategy was recently proposed for the proportional feedback control (PFC) method. Here, it is extended to the highly popular delayed feedback control (DFC) method. Since the DFC method treats the system as a delay-differential equation whose phase space is infinite-dimensional, the characteristic multipliers are found through a truncation in the number of delayed states. Control of a target UPO is achieved for several values of the forcing amplitude. We compare the DFC and PFC methods in terms of stability of the controlled orbit, steady state error and control effort.     

Immunization and Max–Min Optimal Control

A model of a term structure of interest rates is conceived in which disturbances are unknown and bounded. Arbitrage opportunities are ruled out by imposing suitable constraints to the disturbances. This sets the stage for casting a well-known immunization problem as a max-min optimal control problem. Dynamic programming is then used to obtain the analytical solution to such a problem. In this manner, a well-known immunization policy is proved to be optimal in a dynamic setting.     

Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic

We address a rate control problem associated with a single server Markovian queueing system with customer abandonment in heavy traffic. The controller can choose a buffer size for the queueing system and also can dynamically control the service rate (equivalently the arrival rate) depending on the current state of the system. An infinite horizon cost minimization problem is considered here. The cost function includes a penalty for each rejected customer, a control cost related to the adjustment of the service rate and a penalty for each abandoning customer. We obtain an explicit optimal strategy for the limiting diffusion control problem (the Brownian control problem or BCP) which consists of a threshold-type optimal rejection process and a feedback-type optimal drift control. This solution is then used to construct an asymptotically optimal control policy, i.e. an optimal buffer size and an optimal service rate for the queueing system in heavy traffic. The properties of generalized regulator maps and weak convergence techniques are employed to prove the asymptotic optimality of this policy. In addition, we identify the parameter regimes where the infinite buffer size is optimal.     

Optimal recruitment strategies in a multi-level manpower planning model

Markov manpower planning models have extensively been analysed in the past in order to find an optimal personnel strategy for which the stocks of the manpower system evolve towards desirable ones. So far, those models do not take into account interactions among different organizational decision levels. In this paper, a multi-level manpower planning model is presented that considers, besides the desirable stock vector at overall level, proposals for the departmental stocks from lower organizational levels. Attainability of the stock vectors at departmental level is examined under control by recruitment and interdepartmental transitions. A multi-level optimization algorithm is presented to determine an optimal recruitment strategy resulting in attainable and acceptable stocks that are a compromise between the proposal from the top and the proposals from the departments.     

Optimal switching control for microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Hence, a proper feed strategy is required during the process. In this paper, we present an optimal switching control model based on our proposed controlled switching system. Some properties of the controlled switching system are obtained. Subsequently, we prove the existence of optimal control. In order to deduce the optimality conditions, we transcribe the optimal switching control model into an equivalent one with fixed switching instants and parameters. Finally, the optimality conditions of the equivalent problem are investigated by calculus of variations.     

Optimal control problem for a general reaction-diffusion eco-epidemiological model with disease in prey

This paper deals with an optimal control problem for a general reaction-diffusion predator-prey model with disease in prey population. Infected prey will recover from a medication considered as a control strategy. Our primary goal is to characterize an optimal control which minimizes the total density of infected prey and the costs of treatment. Firstly, we obtain the existence and some estimates of the unique strong solution for the controlled system by applying semigroup theory. Subsequently, the existence of optimal pair is proved by means of the technique of minimizing sequence. Furthermore, by proving the differentiability of the control-to-state mapping, we derive the first-order necessary optimality condition, and point out that the optimal is a Bang-Bang control in a special case. Finally, several numerical simulations are performed to illustrate the concrete realization and practical application of the theoretical results obtained in this contribution.     

状态相依效用下的超额损失再保险——投资策略

假设保险公司的盈余过程和金融市场的资产价格过程均由可观测的连续时间马尔科夫链所调节, 以最大化终端财富的状态相依的期望指数效用为目标, 研究了保险公司的超额损失再保险-投资问题. 运用动态规划方法, 得到最优再保险-投资策略的解析解以及最优值函数的半解析式. 最后, 通过数值例子, 分析了模型各参数对最优值函数和最优策略的影响.     

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型;利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指教值说明了控制的有效性.     

Optimal Disturbances of Bistable Time-Delay Systems Modeling Virus Infections

For bistable time-delay dynamical systems modeling the dynamics of viral infections and the virusinduced immune response, an efficient approach is proposed for constructing optimal disturbances of steady states with a high viral load that transfer the system to a state with a low viral load. Functions approximating the behavior of drugs within the framework of well-known pharmacokinetic models are used as basis functions. Optimal disturbances are sought in the W ₂ ¹ norm. It is shown that optimal disturbances found in this norm are superior to those found in the L₂ norm as applied to the development of adequate therapeutic strategies.     

具有年龄结构的捕食-食饵种群动力系统的最优收获控制

研究一类具有年龄结构的捕食-食饵种群动力系统的最优收获策略.由M azur′s定理,证明了最优控制问题解的存在性以及借助于法锥概念得到了最优控制问题解存在的必要条件,推广了已有的结果.     

Optimal Robust Control of Robotic Systems

There are continuously increasing demands to robotic systems (RS) for faster operation, higher accuracy, and lower energy cost. For the purpose of feedback stabilization, we design decentralized controllers with maximum degree of robustness. The control design criterion is based on a generalized diagonal dominant (GDD) condition proven to be necessary and su.cient for such controllers to be robust against arbitrary, but bounded, disturbances. It is defined by optimal trade‐off relations between the bounds of disturbances, the required accuracy, and the control force limits.     

Optimal control in a macroeconomic problem

The Pontryagin maximum principle is used to develop an original algorithm for finding an optimal control in a macroeconomic problem. Numerical results are presented for the optimal control and optimal trajectory of the development of a regional economic system. For an optimal control satisfying a certain constraint, an invariant of a macroeconomic system is derived.