Solving multi‐objective dynamic optimization problems with fuzzy satisfying method

This article proposes a novel algorithm integrating iterative dynamic programming and fuzzy aggregation to solve multi‐objective optimal control problems. First, the optimal control policies involving these objectives are sequentially determined. A payoff table is then established by applying each optimal policy in series to evaluate these multiple objectives. Considering the imprecise nature of decision‐maker's judgment, these multiple objectives are viewed as fuzzy variables. Simple monotonic increasing or decreasing membership functions are then defined for degrees of satisfaction for these linguistic objective functions. The optimal control policy is finally searched by maximizing the aggregated fuzzy decision values. The proposed method is rather easy to implement. Two chemical processes, Nylon 6 batch polymerization and Penicillin G fed‐batch fermentation, are used to demonstrate that the method has a significant potential to solve real industrial problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Guaranteed cost solution of optimal control and game problems for uncertain systems

A class of systems having white noise parameter variations is considered. Solutions of the infinite-time quadratic optimal control (LQOC) and differential game (LQDC) problems are considered. It is shown that guaranteed cost solutions of these problems are possible if the effects of parameter uncertainties are first taken into account in a specified manner. The method is illustrated through a numerical example.     

Multi-objective optimization for fuzzy controller in neuro-muscular rehabilitation

The multi-objective optimization problem could be generally formulated as minimization of vector objectives subject to a number of constraints and bounds. Among the non-derivative problems, genetic algorithms have been proved to be a good solution in addressing ill-posed problem. We applied the multi-objective optimization to fuzzy controller design in order to obtain a optimal solution that is often a compromise between different objectives. The application that are targeted in this paper is the e-health in neuromuscular rehabilitation using electromyographies. Because our application is implemented using a micro-controller with limited capabilities, that is flash memory, computational effort, and the clock frequency, the interest is to design the smallest fuzzy architecture possible in order to accomplish the objectives. The optimal set was used in order to achieves these objectives.     

A fractional Fokker–Planck control framework for subdiffusion processes

An efficient framework for the optimal control of the probability density function of a subdiffusion process is presented. This framework is based on a fractional Fokker–Planck equation that governs the time evolution of the PDF of the subdiffusion process and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open‐loop optimality systems in a model predictive control strategy. The resulting optimality system with fractional evolution operators is discretized by a suitable scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

A framework for robust quadratic optimal control with parametric dynamic model uncertainty using polynomial chaos

We propose a framework tailored to robust optimal control (OC) problems subject to parametric model uncertainty of system dynamics. First, we identify a generic class of robust objective kernels that are based on the class of deterministic quadratic objectives. It is demonstrated how such kernels can be expressed as a function of the stochastic moments of the state and how the objective terms relate to the robustness and performance of the optimal solution. Second, we engage the generalized polynomial chaos (gPC) framework to propagate uncertainty along the state trajectory. Integrating both frameworks makes way to reformulate the problem as a deterministic OC problem in function of the gPC expansion coefficients that can be solved using existing methods. We apply the framework to solve the problem of robust optimal startup behavior of a nonlinear mechanical drivetrain that is subject to a bifurcation in its dynamics.     

Combination of optimal control approaches for aircraft conflict avoidance via velocity regulation

This paper deals with optimal control applied to one of the most crucial and challenging problems in Air Traffic Management that of aircraft conflict avoidance. We propose an optimal control model where aircraft separation is achieved by changing the speeds of aircraft, and the integral over a time window of their squared accelerations is minimized. Pairwise aircraft separation constraints constitute the main difficulties to be handled. We propose an original decomposition of the problem into 3 zones, in such a way that in two of them, no conflict occurs. Then, using the Pontryagin maximum principle, 2 new formulations of the original optimal control problem are proposed and solved via direct shooting methods. Thanks to our decomposition, these numerical methods are applied on subproblems having reduced size with respect to the original one, thus improving the efficiency of the solution process. Thirty problem instances are numerically solved, showing the effectiveness of the proposed approaches.     

Existence and stability analysis of optimal control

By constructing a complete metric space and a compact set of admissible control functions, this paper investigates the existence and stability of solutions of optimal control problems with respect to the right‐hand side functions. On the basis of set‐valued mapping theory, by introducing the notion of essential solutions for optimal control problems, some sufficient and necessary criteria guaranteeing the existence and stability of solutions are established. New derived criteria show that the optimal control problems whose solutions are all essential form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem. The example shows that not all optimal control problems are stable. However, our main result shows that, in the sense of Baire category, most of the optimal control problems are stable. Copyright © 2013 John Wiley & Sons, Ltd.     

An extended penalty function approach to the numerical solution of constrained optimal control problems

This paper presents the extended penalty function method for solving constrained optimal control problems. Here, equality and inequality constraints on the state and control variables are considered. Using the extended penalty function method, the original constrained optimal control problem is transformed into a sequence of optimal control problems without inequality constraints. This is accomplished by adding to the cost functional a penalty term that takes on large values when the inequality constraints are violated and small values when the constraints are satisfied. Also presented is a continuation method for solving the sequence of differential-algebraic boundary value problems arising from the transformed optimal control problems. The effectiveness of the approach is demonstrated via examples.     

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.     

A class of multiobjective control problems

We introduce the class of MP‐pseudoinvex multiobjective optimal control problems. We show that the concept of MP‐pseudoinvexity is a sufficient condition of optimality and, further, that problems such that every control process satisfying Pontryagin's maximum principle is an optimal process are necessarily MP‐pseudoinvex problems. Moreover, a sub‐class of the MP‐pseudoinvex problems, which we call MP‐invex multiobjective optimal control problems, is defined. We prove that the set of optimal solutions of MP‐invex multiobjective problems coincides with the set of optimal solutions of a related scalar problem. Copyright © 2008 John Wiley & Sons, Ltd.     

Applications of the exterior penalty method in constrained optimal control problems

This paper gives a theoretical analysis of the applications of the exterior penalty method in continuous-time non-linear programs and constrained optimal control problems. As an example, the Cauchy Inequality in the continuous case is proved. Also, the exterior penalty method is used to treat constrained optimal control problems. It is proved, under suitable assumptions, that a constrained optimal control problem can be solved by performing a sequence of unconstrained optimal control problems, and the constrained solution to the constrained optimal control problem can be obtained as the limit of the solutions to the sequence of unconstrained optimal control problems. In using the exterior penalty method to solve constrained optimal control problems it is usually assumed that each of the modified unconstrained optimal control problems has at least one solution. An existence theorem for these problems is also given.     

The infinite-order singular problem

In singular optimal control problems, the optimal control is determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. In certain classes of problems, the control never appears, and such problems are termed infinite-order singular problems. It is shown that this class has many useful properties with respect to the theory and computation of optimal controls. In particular, it is shown that for the time-invariant, singular, linear-quadratic problem: (i) the singular order is infinity or less than or equal to the state dimension, (ii) infinite-order problems can arise only from exact differential type cost functions, (iii) the range of the second-variation operator (Hessian) is finite-dimensional, (iv) the computational method converges strongly, and (v) conjugate direction methods converge in a finite number of steps. The latter property is especially useful in the generation of test problems for optimal control computation schemes.     

Structure‐preserving local optimal control of mechanical systems

While system dynamics are usually derived in continuous time, respective model‐based optimal control problems can only be solved numerically, ie, as discrete‐time approximations. Thus, the performance of control methods depends on the choice of numerical integration scheme. In this paper, we present a first‐order discretization of linear quadratic optimal control problems for mechanical systems that is structure preserving and hence preferable to standard methods. Our approach is based on symplectic integration schemes and thereby inherits structure from the original continuous‐time problem. Starting from a symplectic discretization of the system dynamics, modified discrete‐time Riccati equations are derived, which preserve the Hamiltonian structure of optimal control problems in addition to the mechanical structure of the control system. The method is extended to optimal tracking problems for nonlinear mechanical systems and evaluated in several numerical examples. Compared to standard discretization, it improves the approximation quality by orders of magnitude. This enables low‐bandwidth control and sensing in real‐time autonomous control applications.     

A generalized Nash equilibrium approach for optimal control problems of autonomous cars

We consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models.     

周目标教学在护理管理临床实习带教中的应用

目的 探讨周目标教学在护理管理临床实习带教中的应用效果.方法 将2008～2009年实习的76名学生按时间分为对照组和实验组各38名.对照组按常规带教方法,实验组按临床实习要求制定周目标后,带教教师和护士长按照周目标实习要求,指导学生实习并认真管理.结果 两组学生临床能力及学生对教师的满意度比较,差异有统计学意义(均P...     

Method for solving chance constrained optimal control problems using biased kernel density estimators

A method is developed to numerically solve chance constrained optimal control problems. The chance constraints are reformulated as nonlinear constraints that retain the probability properties of the original constraint. The reformulation transforms the chance constrained optimal control problem into a deterministic optimal control problem that can be solved numerically. The new method developed in this paper approximates the chance constraints using Markov Chain Monte Carlo sampling and kernel density estimators whose kernels have integral functions that bound the indicator function. The nonlinear constraints resulting from the application of kernel density estimators are designed with bounds that do not violate the bounds of the original chance constraint. The method is tested on a nontrivial chance constrained modification of a soft lunar landing optimal control problem and the results are compared with results obtained using a conservative deterministic formulation of the optimal control problem. Additionally, the method is tested on a complex chance constrained unmanned aerial vehicle problem. The results show that this new method can be used to reliably solve chance constrained optimal control problems.     

Geostationary debris mitigation using minimum time solar sail trajectories with eclipse constraints

Minimum time solar sailing trajectories are introduced using a combination of indirect and direct optimal control techniques. Here, large‐scale, multiphase optimal control problems are solved using a pseudospectral collocation technique applied to an orbital debris mitigation concept. These solutions are obtained for realistic sail dimensions, producing multirevolution, Earth‐centered trajectories while accounting for uncontrolled spacecraft dynamics in the eclipse regions. Specifically, minimum time solutions for orbit transfer and phasing maneuvers are obtained using only solar radiation pressure for propulsion and control. First, an optimal primer vector steering history is obtained through numerical approximation. Locally optimal, closed‐form solutions are then implemented based on the primer vector direction, resulting in minimum time satisfaction of desired terminal orbital conditions. In addition, a novel solution strategy is presented for multiphase optimal control problems characterized by uncontrollable dynamics with flexible phase boundaries. The maneuvers presented will be shown to enable efficient orbital debris mitigation for large‐scale debris in geostationary orbits.     

Pseudospectral method for fractional infinite horizon optimal control problems

Up to now, several numerical methods have been presented to solve finite horizon fractional optimal control problems by researchers, while solving fractional optimal control problems on infinite domain is a challenging work. Hence, in this article, a numerical method is proposed to solve fractional infinite horizon optimal control problems. At the first stage, a domain transformation technique is used to map the infinite domain to a finite horizon. Also, fractional derivative defined on an unbounded domain is converted into an equivalent derivative on a finite domain. Then, a new shifted Legendre pseudospectral method is utilized to solve the obtained finite problem and a nonlinear programming problem is suggested to approximate the optimal solutions. Finally, some numerical examples are given to show the efficiency of the method.     

Asymptomatic carriers in transmission dynamics of dengue with control interventions

In this paper, we introduce a dengue disease transmission model with symptomatic and asymptomatic infectious classes. We obtain the threshold dynamics governed by the basic reproduction number of the system. We obtain sensitivity analysis of threshold dynamics to recognize the dominant factors that seriously affect the dengue infection. It has been shown that the biting rate and the rate of asymptomatic cases are more sensitive to the basic reproduction number, and predicts that control of mosquito size plays an essential role in reducing equilibrium level of dengue infection. Hence, the use of mosquito nets and control of population size of mosquitoes are highly suggested to the public. We use optimal control theory to help the public health personnel and biologists to adopt better understanding of the modeling strategies to control dengue fever. We apply preventive control, treatment, and insecticide spray to reach the desire objectives; moreover, the existence of the proposed optimal control problem is established analytically and achieves necessary conditions for optimal controls. The simulations obtained suggested that control measures such as mosquito eradication and preventive strategies effectively eradicate and control dengue infections during the epidemic.     

A computational method for a general class of optimal control problems involving integrodifferential equations

In this paper we consider a general class of optimal control problems involving integrodifferential equations. The integral equation component is a Volterra integral equation with convolution kernel. A method is proposed to approximate the kernel which gives rise to a system of ordinary differential equations approximating the Volterra integral equation. Thus the optimal control problem is approximated by a sequence of standard optimal control problems involving only differential equations. Each of the approximate problems is solvable by any of the existing methods for standard optimal control problems, such as the gradient restoration algorithm and the control parametrization algorithm. Convergence analysis is also carried out to justify the proposed approximation. As an application we consider an optimal control problem involving production and marketing systems with distributed time lags. This problem is then solved numerically using the proposed method.