Consistent approximation of a nonlinear optimal control problem with uncertain parameters

This paper focuses on a non-standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain. The research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non-evading moving targets. In this paper, we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem. The framework is proved to produce a zeroth-order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem. In addition, we demonstrate the convergence of the corresponding adjoint variables. The accumulation points of a sequence of optimal state-adjoint pairs for the approximate problem satisfy a necessary condition of Pontryagin Minimum Principle type, which facilitates assessment of the optimality of numerical solutions.     

An optimal control methodology for plant growth—Case study of a water supply problem of sunflower

An optimal control methodology is proposed for plant growth. This methodology is demonstrated by solving a water supply problem for optimal sunflower fruit filling. The functional–structural sunflower growth is described by a dynamical system given soil water conditions. Numerical solutions are obtained through an iterative optimization procedure, in which the gradients of the objective function, i.e. the sunflower fruit weight, are calculated efficiently either with adjoint modeling or by differentiation algorithms. Further improvements in sunflower yield have been found compared to those obtained using genetic algorithms in our previous studies. The optimal water supplies adapt to the fruit filling. For instance, during the mid-season growth, the supply frequency condenses and the supply amplitude peaks. By contrast, much less supplies are needed during the early and ending growth stages. The supply frequency is a determining factor, whereas the sunflower growth is less sensitive to the time and amount of one specific irrigation. These optimization results agree with common qualitative agronomic practices. Moreover they provide more precise quantitative control for sunflower growth.     

Consideration of plant behaviour in optimal servo-compensator design

Where the most prevalent optimal servo-compensator formulations penalise the behaviour of an error system, this paper considers the problem of additionally penalising the actual states and inputs of the plant. Doing so has the advantage of enabling the penalty function to better resemble an economic cost. This is especially true of problems where control effort needs to be sensibly allocated across weakly redundant inputs or where one wishes to use penalties to soft-constrain certain states or inputs. It is shown that, although the resulting cost function grows unbounded as its horizon approaches infinity, it is possible to formulate an equivalent optimisation problem with a bounded cost. The resulting optimisation problem is similar to those in earlier studies but has an additional ‘correction term’ in the cost function, and a set of equality constraints that arise when there are redundant inputs. A numerical approach to solve the resulting optimisation problem is presented, followed by simulations on a micro–macro positioner that illustrate the benefits of the proposed servo-compensator design approach.     

Joint production,inspection and maintenance control policies for deteriorating system under quality constraint

This paper studies the integration of production, sampling inspection and age-based maintenance planning for an unreliable production system subject to gradual deterioration. The deterioration process of the production unit has a twofold effect on its reliability and product quality. To mitigate the effects of such deterioration, an age-based major maintenance can be conducted, which denotes a perfect repair that restores the production unit to initial conditions. The quality control is performed through a sampling plan that inspects a fraction of the parts produced. The problem further considers that the optimal decision must be determined under a constraint on the outgoing quality required by the final customer. In this domain, standard sampling procedures are applicable only to production process that are statistically stable and under control. Nevertheless, such sampling plans disregard the interaction with production management and maintenance issues and they do not consider the effects of deterioration. In this paper a new joint control policy considering the interactions between production-quality and maintenance is proposed. A stochastic mathematical model is developed through specialized optimization techniques to solve such quality constrained problem. Numerical examples are provided to illustrate the usefulness of the proposed approach and to study the interactions between production-quality and maintenance strategies. An extensive sensitivity analysis and a comparative study are conducted to illustrate the effectiveness of the obtained joint control policy.     

A fuzzy control system with application to production planning problems

A considerable part of the literature on fuzzy sets is devoted to the field of fuzzy control system. In this paper, an alternative control system is introduced to describe a dynamic system with fuzzy white noise. In order to find optimal ways to control such a system, fuzzy optimal control theory is further developed. Specifically, a linear quadratic model is formulated and solved as a fuzzy optimal control problem. The formulation and solution of this model provide an economic interpretation of a production planning model both in the finite horizon and in the infinite horizon.     

Infinite horizon optimal control of linear discrete time systems with stochastic parameters

The infinite horizon optimal control problem is considered in the general case of linear discrete time systems and quadratic criteria, both with stochastic parameters which are independent with respect to time. A stronger stabilizability property and a weaker observability property than usual for deterministic systems are introduced. It is shown that the infinite horizon problem has a solution if the system has the first property. If in addition the problem has the second property the solution is unique and the control system is stable in the mean square sense. A simple necessary and sufficient condition, explicit in the system matrices, is given for the system to have the stronger stabilizability property. This condition also holds for deterministic systems to be stabilizable in the usual sense. The stronger stabilizability and weaker observability properties coincide with the usual ones if the parameters are deterministic.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

Chaos, estimation and optimal control of habitat destruction model with uncertain parameters

The present paper discusses chaos, estimation and optimal control of the habitat destruction model with unknown parameters. The linear stability analysis of the steady states of the model will be discussed. Further, the chaotic behavior of this system will be investigated. The dynamic estimators of the unknown parameters and their updating rules are derived. Using Pontryagin principle, the optimal control inputs are derived with respect to a selected measure. Finally, a numerical simulation study for various parameters and different initial densities is presented.     

Equivalent optimal compensation problem in the delta domain for systems with white stochastic parameters

The equivalent representation in the delta domain of the digital optimal compensation problem is provided and computed in this article. This problem concerns finding digital optimal full and reduced-order output feedback controllers for linear time-varying and time-invariant systems with white stochastic parameters. It can subsequently be solved in the delta domain using the strengthened optimal projection equations that we recently formulated in this domain as well. If the sampling rate becomes high, stating and solving the problem in the delta domain becomes necessary because the conventional discrete-time problem formulation and solution become ill-conditioned. In this article, by means of several numerical examples and compensator implementations, we demonstrate this phenomenon. To compute and quantify the improved performance when the sampling rate becomes high, a new delta-domain algorithm is developed. This algorithm computes the performance of arbitrary digital compensators for linear systems with white stochastic parameters. The principle application concerns nonconservative robust digital perturbation feedback control of nonlinear systems with high sampling rates.     

A formula for the optimal cost in the general discrete-time LEQG problem

This paper is aimed at deriving an explicit formula for the optimal cost for discrete-time linear exponential-of-quadratic Gaussian (LEQG) control problems. We make direct calculations for the general case with cross terms in the cost and noise covariance matrices using an information-state approach.     

Numerical solution of optimal control problem for a model of tumour growth with drug application

In this study, a combination of spectral and fixed point methods is used to solve an optimal control problem for a model of tumour growth. The growth of tumour is modelled using three first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the optimal control problem, four control variables are employed to control the concentration of nutrient and drug on the boundary and inside the tumour. Since the problem is nonlinear, applying the fixed point method, in each step of iteration, the problem is changed to a linear one and the parabolic equations are solved using the spectral method. The convergence and stability of method are proven. Some examples are considered to illustrate the efficiency of method. Finally, some figures are provided to reflect the effects of control on the densities of tumourcells.     

Hybrid method for a general optimal sensor scheduling problem in discrete time

In this paper, we consider a general class of optimal sensor scheduling problems in discrete time. There are N₁ sensors available for acquiring data so as to estimate the needed but unknown signal. Only N₂ out of the N₁ sensors can be turned on at any moment, while different weights can be assigned to different sensors. This problem is formulated as a discrete time deterministic optimal control problem involving both discrete and continuous valued controls. A computational method is developed for solving this discrete time deterministic optimal control problem based on a branch and bound method in conjunction with a gradient-based method. The branch and bound method is used to determine the optimal schedule of sensors, where a sequence of lower bound dynamic systems is introduced so as to provide effective lower bounds for the construction of the branching rules. Each of the branches is an optimal weight vector assignment problem and a gradient-based method is developed for solving this optimal control problem. For illustration, two numerical examples are solved.     

An optimal stopping time problem with time average cost in a bounded interval

An optimal stopping time problem for a diffusion process stopped at the boundary of a bounded interval is studied. Our cost functional is of time average type which is different from standard long term average cost functional but can be considered as a version of the Gittins index. Both characterization of the, value function and construction of optimal stopping policy will be presented.     

Optimal placement of active bars in active vibration control for piezoelectric intelligent truss structures with random parameters

Considering the randomness of physical parameters of structural material, geometric dimensions of active bars and passive bars, applied loads and control forces simultaneously, the optimization of active bar’s placement and feedback gains for the vibration control of intelligent truss structures are studied in this paper. Firstly, the performance function is developed based on the maximization of dissipation energy due to control action. Then, the optimal mathematical model with the reliability constraints on dynamic stress and displacement response is built. The numerical feature of dynamic response based on probability of intelligent structure is developed. Finally, a planar intelligent truss structure is used as an example to demonstrate the rationality and validity of the presented model and approach in structural active vibration control.     

Direct scheme in optimal control problems with fast and slow motions

The nonlinear optimal control problems described by singularly perturbed models are studied in order to obtain the ways of decomposition and to construct the suboptimal control sequences. A new scheme, the so-called direct scheme, applying the Vasil'eva boundary layer functions method in control theory is introduced. It consists in directly expanding the problem's conditions into postulated asymptotic series and then in successively solving lower-dimensional approximate decomposition problems of optimal control. By means of the direct scheme a new property of an asymptotic expansion — the relaxation — is obtained, i.e. the decrease of the performance index with each new control approximation. Illustrating examples are given.     

Performance bounds in linear control of unstable MIMO systems with pole location constraint

This paper proposes a methodology to compute quadratic performance bounds when the closed loop poles of a discrete-time multivariable control loop are confined to a disk, centred at the origin, and with radius less than one. The underlying philosophy in this constraint is to avoid certain undesirable dynamic features which arise in quadratic optimal designs. An expression for the performance loss due to the pole location constraint is also provided. Using numerical examples, we show that the performance loss is compensated by an improved transient performance, specially visible in the control signals.     

Solving the Beck and Wieland model with optimal experimentation in DualPC

Currently, there is a renewed interest in the use of optimal experimentation (adaptive control) in economics. Example are found in [Amman and Kendrick, 1999], [Amman and Kendrick, 2003], [Cosimano, in?press], [Cosimano and Gapen, 2005b], [Cosimano and Gapen, 2005a], [Cosimano and Gapen, 2006], [Tesfaselassie et?al., 2007], [Tucci, 1997], [Wieland, 2000a] and [Wieland, 2000b]. In this paper we present the Beck & Wieland model [Beck, G., & Wieland, V. (2002). Learning and control in a changing economic environment. Journal of Economic Dynamics and Control, 26, 1359-1378] and the methodology to solve this model with time-varying parameters using the various control methods described in [Kendrick, 1981] and [Kendrick, 2002]. Furthermore, we also provide numerical results using the DualPC software [Amman, H. M., & Kendrick, D. A. (1999). The DualI/DualPC software for optimal control models: User’s guide. Working paper, Austin, TX 78712, USA: Center for Applied Research in Economics, University of Texas] and show first evidence that optimal experimentation or Dual Control may produce better results than Expected Optimal Feedback.     

Fractional order fuzzy-PID control of a combined cycle power plant using Particle Swarm Optimization algorithm with an improved dynamic parameters selection

The effectiveness of the Particle Swarm Optimization (PSO) algorithm in solving any optimization problem is highly dependent on the right selection of tuning parameters. A better control parameter improves the flexibility and robustness of the algorithm. In this paper, a new PSO algorithm based on dynamic control parameters selection is presented in order to further enhance the algorithm's rate of convergence and the minimization of the fitness function. The powerful Dynamic PSO (DPSO) uses a new mechanism to dynamically select the best performing combinations of acceleration coefficients, inertia weight, and population size. A fractional order fuzzy-PID (fuzzy-FOPID) controller based on the DPSO algorithm is proposed to perform the optimization task of the controller gains and improve the performance of a single-shaft Combined Cycle Power Plant (CCPP). The proposed controller is used in speed control loop to improve the response during frequency drop or change in loading. The performance of the fuzzy-FOPID based DPSO is compared with those of the conventional PSO, Comprehensive Learning PSO (CLPSO), Heterogeneous CLPSO (HCLPSO), Genetic Algorithm (GA), Differential Evolution (DE), and Artificial Bee Colony (ABC) algorithm. The simulation results show the effectiveness and performance of the proposed method for frequency drop or change in loading.