Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

Optimization of pulse frequency modulated control systems via modified maximum principle

The problem of finding the optimum control function for the pulse frequency modulated (PFM) system is considered in this paper. In the PFM systems discussed here, the control function consists of a series of standard pulses. The optimization procedure consists of determining the polarity and positions of the pulses which make up the control function. The performance index is assumed to be a linear combination of the final values of the state variables. This does not exclude the problem of optimizing a system with respect to an integral, provided that the integrand is linear with respect to the state variables, but not necessarily with respect to the control function. In the PFM systems considered, the control function is fixed for a period of time following the initiation of each pulse. This fact precludes the direct application of the existing standard optimizing techniques. The Modified Maximum Principle is presented. It is based on Pontryagin's Maximum Principle and is applicable to open-loop systems with linear plants with fixed operating time. The Modified Maximum Principle is valid for systems with and without final value constraints on the state variables.     

Optimal control problem in bond graph formalism

This paper presents a new way to derive an optimal control system for a specific optimisation problem, based on bond graph formalism. The procedure proposed concerns the optimal control of linear time invariant MIMO systems and can deal with both cases of the integral performance index, these correspond to dissipative energy minimization and output error minimization. An augmented bond graph model is obtained starting from the bond graph model of the system associated with the optimal control problem. This augmented bond graph, consisting of the original model representation coupled to an optimizing bond graph, supplies, by its bicausal exploitation, the set of differential-algebraic equations that analytically give the solution to the optimal control problem without the need to develop the analytical steps of Pontryagin’s method. The proof uses the Pontryagin Maximum Principle applied to the port-Hamiltonian formulation of the system.     

Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems

To partially implement the idea of considering nonlinear optimal control problems immediately on the set of Pontryagin extremals (or on quasiextremals if the optimal solution does not exist), we introduce auxiliary functions of canonical variables, which we call bipositional, and the corresponding modified Lagrangian for the problem. The Lagrangian is subject to minimization on the trajectories of the canonical system from the Maximum Principle. This general approach is further specialized for nonconvex problems that are linear in state, leading to a nonstandard dual optimal control problem on the trajectories of the adjoint system. Applying the feedback minimum principle to both original and dual problems, we have obtained a pair of necessary optimality conditions that significantly strengthen the Maximum Principle and admit a constructive realization in the form of an iterative problem solving procedure. The general approach, optimality features, and the iterative solution procedure are illustrated by a series of examples.     

On smooth optimal control determination

When using the Pontryagin Maximum Principle in optimal control problems the most difficult part of the numerical solution is associated with the non-linear operation of the maximization of the Hamiltonian over the control variables. For a class of problems, the optimal control vector is a vector function with continuous time derivatives. A method is presented to find this smooth control without the maximization of Hamiltonian. Three illustrative examples are considered.     

Velocity profile optimization of on road vehicles: Pontryagin's Maximum Principle based approach

Driving profile of on road vehicles has shown to have significant effect on fuel economy. This paper discusses the development of Pontryagin's Maximum Principle (PMP) based solution to determine the energy optimal velocity profile by incorporating the gear shifting, speed limit and road grade constraints simultaneously. In the proposed approach the real world road grade profile and speed limits are approximated by a set of piece-wise constant functions and the corresponding first order necessary conditions are derived. By solving a number of differential equations an analytical solution is generated. Therefore, the computation time of the solution is extremely fast. To verify the global optimality of the solution, the results are compared with dynamic programming (DP) solution that solves the complex and non-linear representative model of the actual test vehicle. The comparison results prove that the generated optimal speed trajectories are very close to global optimal solution.     

Time optimal control of second-order systems with transport lag†

The time optimal control of particular second–order systems with constant transport lags is determined by application of Pontryagin's Maximum Principle. A system with a delayed control and a system with a delayed state are considered. The optimal control for these systems is compared with the optimal control of a similar system which does not contain transport lag. It is found that the optimal control is essentially bang–bang but that it is non–unique under certain conditions.     

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.     

Mathematical solution of the problem of optimal control of integrated power systems with generalized maximum principle†

This paper deals with the optimal control of an integrated (hydro-thermal) power system taking into account all the non-linearities, constraints and discontinuities in the system. The mathematical solution of the deterministic problem is obtained using the generalized Maximum Principle. It turns out that the optimization equations for the thermal power system obtained by Carpentier and Sirioux, who used the Kuhn and Tucker theory, are a particular case of the optimization equations for the integrated power system obtained here using the generalized maximum principle.     

Optimization of the separation of two species in a chemostat

In this work, we study a two species chemostat model with one limiting substrate, and our aim is to optimize the selection of the species of interest. More precisely, the objective is to find an optimal feeding strategy in order to reach in minimal time a target where the concentration of the first species is significantly larger than the concentration of the other one. Thanks to the Pontryagin Maximum Principle, we introduce a singular feeding strategy which allows to reach the target, and we prove that the feedback control provided by this strategy is optimal whenever initial conditions are chosen in the invariant attractive manifold of the system. The optimal synthesis of the problem in presence of more than one singular arc is also investigated.     

Minimal time bioremediation of natural water resources

We study minimal time strategies for the treatment of pollution in large water volumes, such as lakes or natural reservoirs, with the help of an autonomous bioreactor. The control consists of feeding the bioreactor from the resource, with clean output returning to the resource with the same flow rate. We first characterize the optimal policies among constant and feedback controls under the assumption of a uniform concentration in the resource. In the second part, we study the influence of inhomogeneity in the resource, considering two measurement points. With the help of the Maximum Principle, we show that the optimal control law is non-monotonic and terminates with a constant phase, in contrast to the homogeneous case in which the optimal flow rate decreases with time. This study allows decision makers to identify situations in which the benefit of using non-constant flow rates is significant.     

Recursive modeling and control of multi-link manipulators with vacuum grippers

Manipulators equipped with vacuum grippers are a new and flexible element in innovative material-flow solutions. The limited holding forces of vacuum grippers require elaborate strategies of control to prevent the contact between gripper and load from breaking off especially in time optimal motion. Mathematically this can be modeled as a constraint on internal forces of a multi-link manipulator. A Maximum Principle based approach is presented for the accurate solution of the control problem. Not only the equations of motion of the manipulator, but the complete optimal control problem is modeled recursively. By this, the structural properties of the control problem are revealed. Direct access becomes possible to all information necessary to restrict internal forces efficiently.     

Solving a class of nonlinear optimal control problems via he’s variational iteration method

This paper presents an analytical approximate solution for a class of nonlinear quadratic optimal control problems. The proposed method consists of a Variational Iteration Method (VIM) together with a shooting method like procedure, for solving the extreme conditions obtained from the Pontryagin’s Maximum Principle (PMP). This method is applicable for a large class of nonlinear quadratic optimal control problems. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.     

轮胎成型机计算机监控系统软件设计

本文介绍了轮胎成型机计算机控制系统的组成,工作原理以及工艺程序编译软件和监控软件的编制原理及方法.     

Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls

We obtain necessary global optimality conditions for classical optimal control problems based on positional controls. These controls are constructed with classical dynamical programming but with respect to upper (weakly monotone) solutions of the Hamilton-Jacobi equation instead of a Bellman function. We put special emphasis on the positional minimum condition in Pontryagin formalism that significantly strengthens the Maximum Principle for a wide class of problems and can be naturally combined with first order sufficient optimality conditions with linear Krotov’s function. We compare the positional minimum condition with the modified nonsmooth Ka?kosz-Lojasiewicz Maximum Principle. All results are illustrated with specific examples.     

基于Bang-Bang原理的时间最优控制问题求解

时间最优控制是工程实践中经常遇到的一类最优控制问题。对于较简单的时间最优控制问题可以应用古典变分法和庞特里雅金最大值原理进行分析求解。但在实际问题中,能求得解析解的仅是少数。因此,有必要寻求一种能够有效求解时间最优控制问题的数值方法。在分析时间最优控制问题已有求解方法优缺点的基础上,提出基于Bang—Bang原理和参数最优化方法（遗传算法-单纯形法）相结合求解一类仿射系统的时间最优控制问题的方法。对线性阻尼振子问题进行了数值仿真,结果表明该方法效果良好。     

A new approach to nonlinear optimal feedback law

For a fixed-time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations. They are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations. These equations, involving Lie brackets of vector fields, are derived from the Maximum Principle. An application of this approach to robotics is given.     

A new min-max methodology for computing optimised obstacle avoidance steering manoeuvres of ground vehicles

In this paper, a new methodology for computing optimised obstacle avoidance steering manoeuvres for ground vehicles is presented and discussed. Most of the existing methods formulate the obstacle avoidance problem as an optimal control problem which is hard to solve or as a numerical optimisation problem with a large number of unknowns. This method is based on a reformulation of Pontryagin's Maximum Principle and leads to the solution of an adjustable time optimal controller. The control input is significantly simplified and permits its application in a sample and hold sense. Furthermore, with the proposed approach the maximum tyre forces exerted during the manoeuvre are minimised. In this study, it is shown how to ‘warm start’ the proposed algorithm and which constraints to ‘relax’. Numerical examples and benchmark tests illustrate the performance of the proposed controller and compare it with other standard controllers. A sensitivity analysis for different vehicle parameters is performed and finally conclusions are drawn. A significant advantage of the method is the small computational complexity. The overall simplicity of the controller makes it attractive for application on autonomous vehicles.     

Optimal control of a two-class national economic model

In this paper we extend the one-sector national economic system modelled on the basis of the cyclical growth theory developed by Bergstrom (1967) into a two-class model. This is done by combining and extending the model developed by Pasinetti (1962), Hu (1073), Ahmed and Yeo (1970) and Yeo and Teo (1976). The following economic factors—(i) profit margin, (ii) interest rates, (iii) monetary policy, and (iv) rate of increase of wages—arc considered as control variables. Based on this two-class model and its control variables, the objective of the economy is to maximize the sum of total consumption over a given planning period. The Pontryagin Maximum Principle is used to obtain the forms of the optimal controls. For illustrative purposes, numerical examples are solved using the computer programme of Moore and Too (private communication).     

Some second-order necessary conditions in optimal control

In optimal control problems any extremal arc which trivially satisfies the Maximum Principle, that is a first-order control variation produces no change in cost, is called singular. Higher-order conditions are then needed to check the optimality of such arcs. Using the Volterra series associated with the variation of the cost functional gives a new context for analyzing singular optimal control problems. A basic optimality criterion for a fixed terminal time Mayer problem is obtained which allows one to derive the necessary conditions for optimality in terms of Lie brackets of vector fields associated with the dynamics of the problem.