Minimax Optimal Control Problem Research Articles

Minimax optimal control problems for an extensible beam equation with uncertain initial velocity

This paper is devoted to the problem of minimax optimal control problems of an extensible beam equation with distributed controls and initial velocity disturbances (or noises). The existence of optimal solutions for distributed control with fixed disturbance, namely the (P_{v}) problem, and the existence of minimax optimal solutions to (P) problem without restricting the initial disturbance are proved. We derive the necessary optimality conditions for optimal solutions of the (P_{v}) and (P) problems in the form of Pontryagin’s maximum principle.

Bilinear Minimax Optimal Control Problems of the Velocity Term in an Extensible Beam Equation with Rotational Inertia

The author studies the minimax optimal control problem for an extensible beam equation that takes into account rotational inertia effects. A velocity term multiplied by the bilinear control and disturbance functions to construct a minimax optimal control strategy is added. The existence of an optimal pair, meaning optimal control and perturbation, is proved by assuming some conditions on the considered quadratic cost function. The optimal conditions for optimal pairs are provided by the adjoint systems that correspond to some physically meaningful observation cases.

Robust estimation of state vector coordinates in the controlled helicopter motion problem

The problem of finding H∞ – a observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer’s gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.

Robust Precision Position Tracking of Planar Motors Using Min-Max Model Predictive Control

In this article, a min-max model predictive control (MPC) method of planar motors is proposed for the first time to achieve robust precision position tracking, which has a low computational burden and strong capability to deal with the problems of stability, robustness, optimization, and input constraints. A state-space model with a homogeneous state equation is built to describe the dynamics of the time-varying reference trajectory. Combining the state-space model of the reference trajectory and that of the planar motor, an augmented state-space model is established to obtain an error state formulation. Then, using the error state formulation, a min-max optimal control problem subject to the constraints on bounded uncertainty, stability, and control input is developed. Moreover, applying the theory of asymptotically stable invariant ellipsoids and employing the nested invariant ellipsoids, the explicitly linear state-feedback control laws are obtained using a linear-matrix-inequalities based offline control algorithm. Finally, the min-max MPC is applied to a planar motor system developed in the laboratory for an experimentally comparative study. The results demonstrate the effectiveness of the proposed min-max MPC of planar motor for robust precision position tracking applications.

Optimization Problems of Green Energetics

Introduction. The widespread introduction of "green energy", which is based on renewable energy sources (RES), is recognized as the main direction on which the efforts of the world community should be concentrated in the fight against global warming. The complex network structure of the modern global economy requires a deep scientific substantiation of measures for such a transformation of the energy sector and forecasting their long-term consequences, which, in the face of growing uncertainty associated with the emergence of a large number of new systemic risks, is not an easy task. The decarbonization of the energy sector through greater use of renewable energy is thus a complex minimax optimal control problem and requires the creation of appropriate mathematical models for its solution. The article proposes a mathematical model for determining the dynamics of changes in the share of green energy (SGE) in the overall energy balance, which minimizes the growth rate of greenhouse gases (GG) in the atmosphere and maintains a given standard of living for the population. The purpose of the article is to investigate, using mathematical modeling, how a change in the SGE affects the level of decarbonization and the quality of life of the population, to determine the optimal levels of the SGE, at which the level of GG is minimized and energy production and production functions of some sectors of the economy are maximized. Results. A mathematical model has been developed to determine the conditions under which the growth rate of GG concentration in the atmosphere is minimized and a given standard of living of the population is maintained. The model combines in a single structure similarly described sectors of the economy, each of which is considered in terms of productivity levels, the number of jobs and structural disturbances (six-sector Lorentz model with varying coefficients depending on the level of GG. The model allows us to explore how changes in GG levels and the relationship between various sectors of the economy affect the risks of sustainable development and the quality of life of the population. The conditions for the occurrence of turbulent modes of operation are determined, which lead to an increase in the total number of structural disturbances and a decrease in the total level of productivity. Conclusions. The obtained results show that there is a critical level, starting from which the further growth of SGE in the overall energy balance leads to a reduction in the total volume of products created by the energy sector, which negatively affects the quality of life of the population. The development of this work will be aimed at identifying effective trajectories for the transformation of the energy sector in order to minimize GG emissions and structural disturbances and maximize the levels of production functions of economic sectors. A multicriteria optimal control problem will be investigated, which would allow solving practical problems of energy transformation under conditions of increased uncertainty and risks. Keywords: mathematical modeling, green energy, risk management.

Quadratic Cost Minimax Optimal Control Problems for a Semilinear Viscoelastic Equation with Long Memory

In this paper, we study the quadratic cost minimax optimal control problems for a semilinear viscoelastic equation with long memory. A global well-posedness theorem regarding the solutions to its Cauchy problem is given. We formulate the minimax control problem with bilinear control inputs and corresponding disturbances. Under some assumptions, we prove the existence of optimal pairs and give necessary optimality conditions for optimal pairs in some observation cases.

Necessary optimality conditions for minimax optimal control problems with mixed constraints

A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. Illustrative examples are presented.

Minimax Control Problems: Optimality Conditions

The formulation of the minimax control problem is considered. We allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints, under the constraint qualification of Mangassarian-Fromovitz type are discussed. The optimality conditions under a full rank conditions type is derived, and it is shown that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. We also look at the necessary conditions of optimality for simplified versions of the model and degeneracy issues.

Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory

In the present article, we consider a von Kárman equation with long memory. The goal is to study a quadratic cost minimax optimal control problems for the control system governed by the equation. First, we show that the solution map is continuous under a weak assumption on the data. Then, we formulate the minimax optimal control problem. We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.

Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.

Fréchet differentiability for an extensible beam equation and its application to bilinear minimax control problems

We consider an extensible beam equation with Dirichlet boundary conditions. The object is to show the Fréchet differentiability of nonlinear solution map from a bilinear control input to the solution of the beam equation. We used this method to formulate the minimax optimal control problem. We also show the existence of optimal pairs and find their necessary optimality conditions.

Robust model predictive control for greenhouse temperature based on particle swarm optimization

Application of model predictive control (MPC) in horticultural practice requires detailed models. However, even highly sophisticated greenhouse climate models are often known to have unknown dynamics affected by bounded uncertainties. To enforce robustness during the controller design stage, this paper proposes a particle swarm optimization (PSO)-based robust MPC strategy for greenhouse temperature systems. The strategy is based on a nonlinear physical temperature affine model. The robust MPC technique requires online solution of a minimax optimal control problem, which optimizes the tradeoff between set point tracking and cost requirements reduction. The minimax optimization problem is reformulated to a nonlinear programming problem with constraints. PSO is used to solve the reformulated problem and priority ranking of constraint fitness is proposed to guarantee that the constraints are satisfied. The results of simulations performed using the proposed control system show that the controller can effectively achieve the set point in the presence of disturbances and that it offers more suitable control variables, higher control precision, and stronger robustness than the conventional MPC.

An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.

Solving minimax control problems via nonsmooth optimization

We address minimax optimal control problems with linear dynamics. Under convexity assumptions, by using non-smooth optimization techniques, we derive a set of optimality conditions for the continuous-time case. We define an approximated discrete-time problem where analogous conditions hold. One of them allows us to design an easily implementable descent method. We analyze its convergence and we show some preliminary numerical results.

Robust design of sensor fusion problem in discrete time

In this paper, we consider a robust sensor scheduling problem which estimates the state of an uncertain process based on measurements obtained by a given set of noisy sensors, where the measurements of sensors are subject to external interference uncertainties. We formulate this problem into a minimax optimal control problem, which is equivalent to a semi-infinite programming problem with a dynamic system. A discretization method is used to solve this problem, where the computation is very large scale in general. We propose an approximation method to reduce the computational complexity. For illustration, two numerical examples are solved.

Minimax optimal control problem with state constraints

In this article, nondegenerate necessary conditions of optimality are derived and discussed for the so-called “minimax” optimal control problems with state constraints. In this class of problems, the data depends on an unknown parameter that takes values in a given compact set called “uncertainty set”. The solution of the problem is considered for the worst case value of the unknown parameter. The result is obtained in the two stages: after the basic necessary conditions of optimality are derived, additional conditions ensuring their nondegeneracy in the minimax context are obtained.

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.

Min–max optimal control of linear systems with uncertainty and terminal state constraints

In this paper, a class of min–max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied. The main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions. Then, the original min–max optimal control problem is solved via solving a sequence of semi-definite programming problems. An example is presented to illustrate the proposed method.

A dual dynamic programming for minimax optimal control problems governed by parabolic equation

In the article, minimax optimal control problems governed by parabolic equations are considered. We apply a new dual dynamic programming approach to derive sufficient optimality conditions for such problems. The idea is to move all the notions from a state space to a dual space and to obtain a new verification theorem providing the conditions, which should be satisfied by a solution of the dual partial differential equation of dynamic programming. We also give sufficient optimality conditions for the existence of an optimal dual feedback control and some approximation of the problem considered, which seems to be very useful from a practical point of view.

Assessment of the Cell Broadband Engine Architecture as a platform to solve closed-loop optimal control problems

The purpose of the paper is to evaluate the potential of the Cell Broadband Engine Architecture for solving closed-loop optimal control problems. The main part contains the description and pseudo language implementation of two dynamic programming algorithms to solve stochastic discounted and minimax optimal control problems of a Polish Upper Vistula river multi-reservoir water system over the infinite time horizon. OpenMP parallelizing directives were applied to allow for high programmer efficiency as opposed to low level, assembler like Cell programming. In an extensive number of performance tests conducted on different systems this approach proved to be very efficient (reaching 20× speed-up) in comparison with general purpose processors validating the CBEA as an attractive platform for solving optimal control problems.