The inverse lure problem of optimal control

Given the linear system $\text{x}\text{}$ = Ax - bu, y = c^Tx, it is shown that, for a certain non-quadratic cost functional, the optimal control is given by u_opt(x) = h(c^Tx), where the function h(y) must satisfy the conditions ky²?h(y)y>0 for y≠0, h(0) = 0 and existence of h^-1 everywhere. The linear system considered must satisfy the Popov condition 1/k + (1 +?ωβ) G(?ω)>0 for all ω, G(s) being the y(s)/u(s) transfer function.     

The obstacle problem of integro-partial differential equations with applications to stochastic optimal control/stopping problem

This paper is devoted to existence and uniqueness of minimal mild super solutions to the obstacle problem governed by integro-partial differential equations. We first study the well-posedness and local Lipschitz regularity of L^p solutions (p?≥?2) to reflected forward-backward stochastic differential equations (FBSDEs) with jump and lower barrier. Then we show that the solutions to reflected FBSDEs provide a probabilistic representation for the mild super solution via a nonlinear Feynman–Kac formula. Finally, we apply the results to study stochastic optimal control/stopping problems.     

Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems

This paper presents the optimal control problem for a non-linear polynomial system with respect to a Bolza-Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional polynomial-quadratic regulator does not lead to a causal solution and, therefore, fails. Performance of the obtained optimal controller is verified in the illustrative example against the conventional polynomial-quadratic regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.     

Bond graph formulation of an optimal control problem for linear time invariant systems

A recent communication has proposed a conjectural procedure for representing a category of optimal control problems in bond graph language [W. Marquis-Favre, B. Chereji, D. Thomasset, S. Scavarda, Bond graph representation of an optimal control problem: the dc motor example, in: ICBGM’05 International Conference of Bond Graph Modelling and Simulation, New Orleans, USA, January 23-27, 2005, pp. 239-244]. This paper aims at providing a fundamental theory for proving the effectiveness of this procedure. The class of problem that the procedure can deal with has been extended. Its application was formerly restricted to linear time invariant siso system. The systems considered now are linear time invariant mimo systems. The optimization objective is the minimization of dissipation and input. The developments concerning the optimal control problem are based on the Pontryagin maximum principle and the proof of the effectiveness of the procedure makes a broad use of the port-Hamiltonian concept. As a result, the bond graph representation of the given optimization problem enables the analytical system, which provides the optimal solution, to be derived. The work presented in this paper is the first step in research with perspectives towards formulating dynamic optimization problems in bond graph and, towards coupling this formulation with a sizing methodology using bond graph language and a state-space inverse model approach. This sizing methodology, however, is not the topic of this paper and thus is not presented here.     

On the set-valued approach to optimal control of sliding mode processes

This paper is devoted to a theoretic framework for a general optimal control problem (OCP) associated with the classic sliding mode process. The sliding dynamic behavior is interpreted here as a special kind of additional constraints related to the main optimization problem. We are specially interested in the development of some adequate constructive approximations of the original OCPs. The mathematical approach based on the set-valued analysis allows to study the discontinuity of sliding mode dynamics in the abstract setting. Moreover, we also establish some sensitivity properties of the optimal solutions. The obtained results provide an universal analytical tool for the corresponding conceptual approximation schemes related to the original OCPs. The constructive approximations proposed in this paper are numerically stable and can be applied to various classes of optimal control processes governed by the affine control systems.     

Chebyshev series approach to system identification,analysis and optimal control

The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.     

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.     

The maximum entropy approach to record abbreviation for optimal record control

This paper presents an entropy based technique for the abbreviation of text strings for use as a control code. Tests were performed using titles from machine readable bibliographic files. Greater than 94% of the titles have been found to generate a unique seven character code.     

Some numerical tests for an alternative approach to optimal feedback control

We test a recently proposed approach to optimal feedback control of nonlinear systems leading to an iterative descending strategy [24]. We start by discussing the numerical implementation of this strategy, and propose a number of improvements that can speed up the computation process by up to two orders of magnitude. The resulting algorithm is then applied to a series of test problems of increasing complexity. Results seem to show that this can be a promising strategy to bear in mind for more realistic situations.     

A direct method based on the Chebyshev polynomials for a new class of nonlinear variable-order fractional 2D optimal control problems

The main goal of this study is to develop an efficient matrix approach for a new class of nonlinear 2D optimal control problems (OCPs) affected by variable-order fractional dynamical systems. The offered approach is established upon the shifted Chebyshev polynomials (SCPs) and their operational matrices. Through the way, a new operational matrix (OM) of variable-order fractional derivative is derived for the mentioned polynomials.The necessary optimality conditions are reduced to algebraic systems of equations by using the SCPs expansions of the state and control variables, and applying the method of constrained extrema. More precisely, the state and control variables are expanded in components of the SCPs with undetermined coefficients. Then these expansions are substituted in the cost functional and the 2D Gauss-Legendre quadrature rule is utilized to compute the double integral and consequently achieve a nonlinear algebraic equation.After that, the generated OM is employed to extract some algebraic equations from the approximated fractional dynamical system. Finally, the procedure of the constrained extremum is used by coupling the algebraic constraints yielded from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the cost functional by a set of unknown Lagrange multipliers. The method is established for three various types of boundary conditions.The precision of the proposed approach is examined through various types of test examples.Numerical simulations confirm the suggested approach is very accurate to provide satisfactory results.     

Modelling and optimal control of a time-delayed switched system in fed-batch process

The main control goal of the fed-batch process is to maximize the yield of target product as well as to minimize the operation costs simultaneously. Considering the existence of time delay and the switching nature in the fed-batch process, a time-delayed switched system is proposed to formulate the 1,3-propanediol (1,3-PD) production process. Some important properties of the system are also discussed. Taking the switching instants and the terminal time as the control variables, a free terminal time delayed optimal control problem is then presented. Using a time-scaling transformation and parameterizing the switching instants into new parameters, an equivalently optimal control problem is investigated. A numerical solution method is developed to seek the optimal control strategy by the smoothing approximation method and the gradient of the cost functional together with that of the constraints. Numerical results show that the mass of target product per unit time at the terminal time is increased considerably.     

Orthogonal functions approach to optimal control of delay systems with reverse time terms

Using block-pulse functions (BPFs)/shifted Legendre polynomials (SLPs) a unified approach for computing optimal control law of linear time-varying time-delay systems with reverse time terms and quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamical systems are converted into linear algebraic equations by using operational matrices of orthogonal functions (BPFs and SLPs). The problem of finding optimal control law is thus reduced to the problem of solving algebraic equations. One example is included to demonstrate the applicability of the proposed approach.     

Free final time fractional optimal control problems

A formulation and solution scheme of free final time fractional optimal control problems is presented in this paper. The dynamic constraint is described by a fractional differential equation. Performance index considered is a function of both the state and control variables. The necessary conditions of optimality and the transversality condition are obtained using Lagrange multiplier technique. A numerical technique similar to Shooting method is used for solving the optimal conditions. Numerical example is provided to show the effectiveness of the formulation and numerical solution scheme. It is interesting to note that the final time changes with the interchange of the boundary conditions, which does not occur in classical optimal control problems.     

挣得值法在船舶建造费用风险控制中的应用

船舶建造项目由于投资巨大、建造周期长,如果控制不当,很容易产生费用风险问题,因此必须要在建造过程中重视对其进行控制.挣得值法是对建造工程费用风险进行控制的一种有效方法.将该方法应用到船舶建造费用风险控制中,通过计算费用偏差来衡量费用风险的大小,进而找出偏差产生的原因,以达到对其进行控制的目的.     

Necessary conditions for Chebyshev-Bolza optimal control problems

A performance index consisting of a Chebyshev absolute maximum functional plus terminal and integral cost is applied to the optimal control of dynamical systems. First-order necessary conditions are derived for a large class of systems. Utilizing the necessary conditions, analytic examples are worked in demonstrating many of the properties of this class of systems.     

An integrated data-driven Markov parameters sequence identification and adaptive dynamic programming method to design fault-tolerant optimal tracking control for completely unknown model systems

In this paper, an integrated design of data-driven fault-tolerant tracking control is addressed relying on the Markov parameters sequence identification and adaptive dynamic programming techniques. For the unknown model systems, the sequence of Markov parameters together with the covariance of innovation signal is firstly estimated by least square method. After a transformation of value function from stochastic to deterministic, a policy iteration adaptive dynamic programming algorithm is then formulated to find the optimal tracking control law. In order to eliminate the influence of unpredicted faults, an active fault-tolerant supervisory control strategy is further constructed by synthesizing fault detection, isolation, estimation and compensation. All these involved designs are performed in the data-driven manner, and thus avoid the information requirement about system drift dynamics. From the perspective of system operation management, the above integrated control scheme provides a framework to achieve the tracking performance optimization, monitoring and maintaining simultaneously. The effectiveness of these conclusions is finally verified via two case studies.     

Minimax optimal control of linear system with input-dependent uncertainty

In this paper, the quadratic minimax optimal control of linear system with input-dependent uncertainty is studied. We show that it admits a unique solution and can be approximated by a sequence of finite-dimensional minimax optimal parameter selection problems. These finite-dimensional minimax optimal parameter selection problems are further reduced to scalar optimization problems which also admit unique solutions. Thus, the original minimax optimal control problem is solved via solving a sequence of simple scalar optimization problems. Numerical experiments are presented to illustrate the developed method.     

A control problem with structural choices

The case is considered in which, during the operation of an optimal control system, the optimizer, in addition to applying his usual control, may switch structures. Necessary and sufficient conditions are derived and emphasis is placed on the special characteristics of this problem. Continuous and discrete time set-ups are considered and the separation principle is shown not to hold for the linear quadratic case in the presence of noise.     

On optimal location and management of a new industrial plant: Numerical simulation and control

Within the framework of numerical modelling and multi-objective control of partial differential equations, in this work we deal with the problem of determining the optimal location of a new industrial plant. We take into account both economic and ecological objectives, and we look not only for the optimal location of the plant but also for the optimal management of its emissions rate. In order to do this, we introduce a mathematical model (a system of nonlinear parabolic partial differential equations) for the numerical simulation of air pollution. Based on this model, we formulate the problem in the field of multi-objective optimal control from a cooperative viewpoint, recalling the standard concept of Pareto-optimal solution, and pointing out the usefulness of Pareto-optimal frontier in the decision making process. Finally, a numerical algorithm – based on a characteristics/Galerkin discretization of the adjoint model – is proposed, and some numerical results for a hypothetical situation in the region of Galicia (NW Spain) are presented.