A Neural Network Approach for Solving Optimal Control Problems with Inequality Constraints and Some Applications

In this paper, a class of nonlinear optimal control problems with inequality constraints is considered. Based on Karush–Kuhn–Tucker optimality conditions of nonlinear optimization problems and by constructing an error function, we define an unconstrained minimization problem. In the minimization problem, we use trial solutions for the state, Lagrange multipliers, and control functions where these trial solutions are constructed by using two-layered perceptron. We then minimize the error function using a dynamic optimization method where weights and biases associated with all neurons are unknown. The stability and convergence analysis of the dynamic optimization scheme is also studied. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method. We also provide two applicable examples in robotic engineering.     

Approximating the Solution of Optimal Control Problems by Fuzzy Systems

In this paper, the ability of fuzzy systems is used to estimate the solution of crisp optimal control problems. To solve an optimal control problem, first the well-known Euler–Lagrange conditions are obtained and then, the solution of these conditions is approximated by defining a trial solution based on fuzzy systems. The parameters of fuzzy systems are adjusted by an optimization algorithm. Numerical examples and comparisons with exact solutions reveal the capability and accuracy of proposed method.     

Optimality criterion for maximum fundamental frequency of free vibrations of frames including axial and bending effects

An optimality criterion for maximum multiple fundamental frequency of free vibrations for structures of prescribed weights is presented. The criterion includes both axial and bending effects and can be used for analysis of truss, beam and frame structures. The error norm based on the criterion is proposed and used to verify trial designs against the optimum. The accompanying iterative procedure reduces this error norm to zero and drives a trial design to the optimum. The modality of the design at the optimum and the corresponding set of Lagrange multipliers are determined automatically.     

Differential dynamic programming methods for solving bang-bang control problems

Differential dynamic programming is a technique, based on dynamic programming rather than the calculus of variations, for determining the optimal control function of a nonlinear system. Unlike conventional dynamic programming where the optimal cost function is considered globally, differential dynamic programming applies the principle of optimality in the neighborhood of a nominal, possibly nonoptimal, trajectory. This allows the coefficients of a linear or quadratic expansion of the cost function to be computed in reverse time along the trajectory: these coefficients may then be used to yield a new improved trajectory (i.e., the algorithms are of the "successive sweep" type). A class of nonlinear control problems, linear in the control variables, is studied using differential dynamic programming. It is shown that for the free-end-point problem, the first partial derivatives of the optimal cost function are continuous throughout the state space, and the second partial derivatives experience jumps at switch points of the control function. A control problem that has an aualytic solution is used to illustrate these points. The fixed-end-point problem is converted into an equivalent free-end-point problem by adjoining the end-point constraints to the cost functional using Lagrange multipliers: a useful interpretation for Pontryagin's adjoint variables for this type of problem emerges from this treatment. The above results are used to devise new second- and first-order algorithms for determining the optimal bang-bang control by successively improving a nominal guessed control function. The usefulness of the proposed algorithms is illustrated by the computation of a number of control problem examples.     

Mixed coordination method for long-horizon optimal control problems

We present a new approach to solving long-horizon, discrete-time optimal control problems using the mixed coordination method. The idea is to decompose a long-horizon problem into subproblems along the time axis. The requirement that the initial state of a subproblem equal the terminal state of the preceding subproblem is relaxed by using Lagrange multipliers. The Lagrange multipliers and initial state of each subproblem are then selected as high-level variables. The equivalence of the two-level formulation and the original problem is proved for both convex and non-convex cases. The low-level subproblems are solved in parallel using extended differential dynamic programming (DDP). An efficient way to find the gradient and hessian of a low-level objective function with respect to high-level variables is developed. The high-level problem is solved using the modified Newton method. An effective procedure is developed to select initial values of multipliers based on the initial trajectory. The method can convexify the high-level problem while maintaining the separability of an originally non-convex problem. The method performs better and faster than one-level DDP for both convex and non-convex test problems.     

Generalisation of Euler–Lagrange equations to find min–max optimal solution of uncertain systems

A major problem in optimal control theory is existing uncertainty in initial state, cost function, or in state equations. In this paper, a generalised Euler–Lagrange approach to find min–max optimal solution of uncertain systems with uncertain or certain cost function using calculus of variation is considered. The work is based on an admissible system path or a trajectory, which minimises the maximum value of the functional over all uncertainty. First of all, a new form of Euler–Lagrange conditions for uncertain systems is presented. Then several cases are indicated where final condition can be specified or free. Also necessary conditions are introduced to existence of min–max optimal solution of the uncertain systems. Also the method is generalised when the uncertainties are bounded with using Pontryagin's minimum principle. Finally, efficiency of the proposed methods is verified through some examples.     

Evolutionary programming techniques for constrained optimizationproblems

Two evolutionary programming (EP) methods are proposed for handling nonlinear constrained optimization problems. The first, a hybrid EP, is useful when addressing heavily constrained optimization problems both in terms of computational efficiency and solution accuracy. But this method offers an exact solution only if both the mathematical form of the objective function to be minimized/maximized and its gradient are known. The second method, a two-phase EP (TPEP) removes these restrictions. The first phase uses the standard EP, while an EP formulation of the augmented Lagrangian method is employed in the second phase. Through the use of Lagrange multipliers and by gradually placing emphasis on violated constraints in the objective function whenever the best solution does not fulfill the constraints, the trial solutions are driven to the optimal point where all constraints are satisfied. Simulations indicate that the TPEP achieves an exact global solution without gradient information, with less computation time than the other optimization methods studied here, for general constrained optimization problems     

An active set algorithm for tracing parametrized optima

Optimization problems often depend on parameters that define constraints or objective functions. It is often necessary to know the effect of a change in a parameter on the optimum solution. An algorithm is presented here for tracking paths of optimal solutions of inequality constrained nonlinear programming problems as a function of a parameter. The proposed algorithm employs homotopy zero-curve tracing techniques to track segments where the set of active constraints is unchanged. The transition between segments is handled by considering all possible sets of active constraints and eliminating nonoptimal ones based on the signs of the Lagrange multipliers and the derivatives of the optimal solutions with respect to the parameter. A spring-mass problem is used to illustrate all possible kinds of transition events, and the algorithm is applied to a well-known ten-bar truss structural optimization problem.     

轨綫两端均受限制时的最优控制问題

本文用文献[4]中的方法,首先处理了轨线两端均受限制时的快速最优控制问题,得到控制最优性的必要条件以及在某种意义下的充分条件;还得到有关微分方程的边界条件,并说明其几何意义,即贯截条件.此外,又讨论了所用方法中乘子的性质及作用. 对于一般意义下的以及文献[4]中所讨论的最优控制问题,当轨线两端均受限时,也可象此处对快速系统那样进行处理,并得到相应的结果.同时,关于贯截条件及乘子的讨论,也仍然有效. 文中附有二个算例.     

A Hermite‐Lobatto Pseudospectral Method for Optimal Control

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre‐Gauss‐Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over‐constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush‐Kuhn‐Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.     

Evolutionary variational inequality with long-term memory and applications to economic networks

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.     

Multiobjective model predictive control

This paper proposes a novel model predictive control (MPC) scheme based on multiobjective optimization. At each sampling time, the MPC control action is chosen among the set of Pareto optimal solutions based on a time-varying, state-dependent decision criterion. Compared to standard single-objective MPC formulations, such a criterion allows one to take into account several, often irreconcilable, control specifications, such as high bandwidth (closed-loop promptness) when the state vector is far away from the equilibrium and low bandwidth (good noise rejection properties) near the equilibrium. After recasting the optimization problem associated with the multiobjective MPC controller as a multiparametric multiobjective linear or quadratic program, we show that it is possible to compute each Pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular Pareto optimal solution. Furthermore, we provide conditions for selecting Pareto optimal solutions so that the MPC control loop is asymptotically stable, and show the effectiveness of the approach in simulation examples.     

Fractional optimal control problems with both integer-order and Atangana–Baleanu Caputo derivatives

This article considers fractional optimal control problems (FOCPs) including both integer-order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method.     

On nonexistence of boundary arcs in control problems with bounded state variables

When solving optimal control problems with bounded state variables, one must determine whether the optimal trajectory intersects the boundary only at isolated points in time (boundary point) or remains on the boundary for a nonzero length of time (boundary arc). Previously, this determination has been made by trial and error. The task is complicated by the fact that the necessary conditions in common use for these problems assume that the solution has a boundary arc, and can thus yield a boundary arc when the solution has no boundary arc. In this paper the necessary conditions of [1] are used to derive conditions under which the optimal trajectory cannot have a boundary arc. These conditions include the condition for no boundary arcs developed in [1] as a special case. The application of these conditions is illustrated via several examples.     

On the characterization of noninferior solutions of the vector optimization problem

This paper treats three familiar characterizations of noninferior solutions of the vector optimization problem in terms of solutions of (i) the -constraint problem; (ii) the weighting problem; and (iii) the Langrangian problem. Interrelationships among the above characterizations are emphasized by means of a unified treatment of various known results found in the literature. In addition to summarizing existing results, we propose necessary and sufficient conditions for proper noninferiority expressed in terms of the positivity of optimal Lagrange multipliers in the -constraint problem.     

Design of maximum thrust plug nozzles with variable inlet geometry

An optimization analysis is presented for axisymmetric plug nozzles with varible inlet geometry. The analysis is based on the governing gas dynamic relations for a rotational flow of a frozen or equilibrium gas mixture. The problem is formulated to maximize the axial thrust produced by the plug nozzle for a general isoperimetric constraint, such as constant nozzle length or constant nozzle surface area. The effects of base pressure and ambient pressure are included in the thrust expression to be maximized. The governing gas dynamic equations and the differential and integral constraints that the solution must satisfy are incorporated into the formulation by means of Lagrange multiples. The formalism of the calculus of variations is applied to the resulting functional to be maximized. The results of the optimization analysis are a set of partial differential equations for determining the Lagrange multipliers in the region of interest and a set of equations for determining the necessary boundary conditions for the solution. The complete set of equations for the gas dynamic properties and the Lagrange multipliers are system of first order, quasi-linear, non-homogeneous partial differential equations of the hyperbolic type, which can be treated by the method of charac- teristics. The characteristic and compatibility equations for the system are presented. A numerical solution procedure is presented to determine wether or not a given plug nozzle geometry is an optimal solution. An iteration technique is developed which systematically adjusts the plug nozzle geometry until the optimal solution is obtained. Selected parametric studies are presented. These studies illustrate the effect of the specific heat ratio, the design pressure ratio and the base pressure model on the thrust peformance and nozzle geometry of optimal, fixed length, plug nozzles.     

翼伞系统最优归航轨迹设计的敏感度分析方法

本文对三自由度翼伞系统归航轨迹优化问题进行了研究,采用控制变量参数化与时间尺度变换相结合的优化算法对翼伞系统的最优控制问题进行数值求解.该方法是基于灵敏度分析的优化算法,将控制量以及控制量转换时间转化为一系列参数优化问题同时进行求解.仿真结果表明,相对于基于两端边值优化算法而言,灵敏度分析法只需要正向积分进行求解,因而具有计算简单、耗时短等优点,其控制效果良好,距离偏差和方向偏差均满足实际需求,有效地提高了翼伞系统的着陆精度,验证了该优化算法的可行性.     

Dynamic analysis of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load

The dynamic response of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load is studied by using Lagrange equations. In the study, for using the Lagrange equations, trial functions denoting the deflection of the beam and the rotation of the cross-sections are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The effects of the value of the eccentricity of the compressive load, the excitation frequency, the constant velocity of the transverse moving harmonic load and viscous damping of the material of beams are studied in detail. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with exact solutions based on the Euler–Bernoulli beam theory obtained for the special cases of the investigated problem.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

Mortar finite elements for a heat transfer problem on sliding meshes

We consider a heat transfer problem with sliding bodies, where heat is generated on the interface due to friction. Neglecting the mechanical part, we assume that the pressure on the contact interface is a known function. Using mortar techniques with Lagrange multipliers, we show existence and uniqueness of the solution in the continuous setting. Moreover, two different mortar formulations are analyzed, and optimal a priori estimates are provided. Numerical results illustrate the flexibility of the approach. The work was supported by the EU-IHP Breaking Complexity project, CEE HPRN-CT-2002-00286.