Trigonometric approximation of optimal periodic control problems

Trigonometric polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion of the constrained optimal periodic control problem. Several discretized problems using such polynomials are proposed to approximate the problem considered. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given. The application to a class of optimal periodic control problems in chemical engineering is discussed.     

Trigonometric approximation of optimal periodic control problems with variable delays

A trigonometric approximation scheme for optimal periodic control problems with state and control dependent delays is considered. The formulation of approximate problems is here adapted to a specific form of state equation with variable time delays. Sufficient conditions for the sequence of nearly optimal solutions of approximate problems to be a generalized minimizing sequence for the basic problem are given. The application of the method proposed to a periodic control of chemical reactors with recycles and controlled piping is pointed out.     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

Remarks on the paper “Optimal control of a class of linear multivariable systems with integral quadratic energy constraint ”

An optimal periodic control problem for a system described by differential equations is considered. Control units are assumed to generate control actions with the square-integrable derivative. The above problem is approximated by a sequence of discretized problems containing trigonometric polynomials, which approximate the state and control variables, and the functions in the criterion and differential equations. The conditions for a sequence of optimal solutions to discretized problems, which are to be a generalized minimizing sequence for the basic problem, are given. Extensions to more general problem formulations are presented. The possibility of application is illustrated by the example of an optimal periodic control problem for a chemical reactor.     

State-space approach to discrete-time 2-optimal control with a causality constraint

Controller design with a causality constraint arises in periodic or multirate control systems. In this paper complete state-space solutions to the optimal and suboptimal ₂ control problems are developed with a causality constraint on controller feedthrough terms. Explicit formulas for the controllers are given in terms of solutions of two Riccati equations. The results are more implementable than existing frequency-domain solutions.     

离散事件动态系统中的控制综合问题

本文将离散事件动态系统（ＤＥＤＳ）监控方法中的控制综合问题作了系统的分类，得到了六种控制综合问题，并将它们表示成泛函极值问题，讨论了它们的可行解，最优解的存在性，可生解集的结构以及相互之间的关系。     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Difference approximation of optimal periodic control problems

The constrained optimal periodic control problem is approximated by a sequence of discretized problems in which the system of differential equations of the basic continuous problem is replaced by a system of one–step difference equations. Two kinds of approximate optimal controls are derived from the optimal solutions of discretized problems: the first in the form of a step function and the second in the form of a special trigonometric polynomial generated by a positive kernel. Sufficient conditions for approximate solutions to be weakly convergent to the optimal solution of the basic problem are given. Certain improvements in the difference approximation considered are discussed and potential applications given.     

Regularized trigonometric approximation of optimal periodic control problems

The optimal periodic control problem with the main and the additional optimality criterion for a system described by differential equations is investigated. The energy and raw material consumption are considered as the additional criteria. The problem characterized is approximated by a sequence of discretized optimization problems using trigonometric polynomials and penalty functions. Sufficient conditions are given for the convergence of solutions of discretized problems to the regular optimal solution, which minimizes the additional criterion over the set of solutions minimizing the main criterion. Strong convergence in the case where the energy consumption is the additional criterion is proven.     

Trigonometric approximation of optimal periodic control problemsfor systems with inertial controllers

A constrained optimal periodic control (OPC) problem for nonlinear systems with inertial controllers is considered. A sequence of approximate problems containing trigonometric polynomials for approximation of the state, control, and functions in the state equations and in the optimality criterion is formulated. Sufficient conditions for a sequence of nearly optimal solutions of approximate problems to be norm-convergent to the basic problem optimal solution are derived. It is pointed out that the direct approximation approach in the space of state and control combined with the finite-dimensional optimization methods such as the space covering and gradient-type methods makes probable the finding of the global optimum for OPC problems     

Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

Suboptimal control of singularly perturbed systems and periodicoptimization

A traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the fast subsystems considered with frozen slow variables and controls. It is shown that such an approximation is valid if and only if certain families of periodic optimization problems admit steady state solutions. It is also shown how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when approximation by reduced problems is not possible     

Application of the singular control test to periodic control problems of a continuous distillation process

The singular control test is applied to periodic control problems of a continuous distillation process in order to determine if the optimal periodic control is proper. It is also shown that the singular control test is easier to apply than the w-teat for the problems of concern. Some numerical calculations are carried out for seeking the optimal periodic controls.     

Feasibility and optimality in structural design

Some relationships between optimal and feasible structural designs are presented. In many structural design problems, where the objective function is not sensitive to changes in the design variables, the object is to find an improved feasible design rather than the theoretical optimum. Interior optimization methods that converge to the optimum from the interior side of the acceptable domain are most suitable for this purpose. Several methods, intended to introduce improved feasible designs, are discussed in this paper. These methods are particularly useful for problems with a narrow feasible region or in cases where it is necessary to determine whether a feasible region exists. Necessary conditions for feasible solutions are derived and a procedure to convert a nonfeasible design into a feasible one by modifying the preassigned parameters is introduced. It is shown that the optimal value of the modified parameters can often be determined directly. Several examples illustrate the relationships between feasibility and optimality. It is emphasized that optimization methods are most useful in the search for improved feasible solutions for various practical design problems.     

A dynamical trajectory-based methodology for systematically computing multiple optimal solutions of general nonlinear programming problems

In this paper, a novel dynamical trajectory-based methodology is developed for systematically computing multiple local optimal solutions of general nonlinear programming problems with disconnected feasible components satisfying nonlinear equality/inequality constraints. The proposed methodology, deterministic in nature, exploits trajectories of two different nonlinear dynamical systems to find multiple local optimal solutions. The methodology consists of two phases: Phase I starts from an arbitrary (infeasible) initial point and finds systematically multiple or all the disconnected feasible components; Phase II finds an adjacent local optimal solution from a local optimum via a decomposition point, thereby systematically locating multiple local optimal solutions which lie within each feasible component found in Phase I. By alternating between these two phases, the methodology locates multiple or all the local optimal solutions which lie in all the disconnected feasible components. A theoretical foundation for the proposed methodology is also developed. The methodology is illustrated with a numerical example with promising results.     

Trigonometric approximation of optimal periodic control problems for systems with cascade structure

A cascade-structured system controlled by the input of the first subsystem is considered. Trigonometric polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion for the constrained optimal periodic control of the above system. The formulation of approximating problems is adapted to the system structure to obtain particularly good approximating properties. Sufficient conditions for the convergence of solutions of discretized problems to the optimal solution of the basic problem are given, and the convergence rate is estimated. An application of the proposed approximation to multi-stage chemical processes is discussed.     

Approximate discrete maximum principle for the difference approximation of optimal periodic control problems

Distinctive features of the difference approximation for optimal periodic control (OPC) problems are considered. It is shown that the convergence conditions on approximate solutions imply consistency conditions on the time grid step and the accuracy of the periodicity constraint approximation such that the approximate discrete maximum principle holds for a sequence of discretized problems. A specific form of this principle when applied to the difference approximation method for OPC problems is analysed and computational methods are discussed. Application of this approach to a class of systems non-linearly dependent on control variables, and occurring in the chemical industry, is suggested.     

Satisficing solutions of multi-objective fuzzy optimization problems using genetic algorithm

In the present paper, a genetic algorithm for multi-objective optimization problems with max-product fuzzy relation equations as constraints is presented. Since the non-empty feasible domain of such problems is, in general, a non-convex set; the traditional optimization methods cannot be applied. Here, we are presenting a genetic algorithm (GA) to find “Pareto optimal solutions” for solving such problems observing the role of non-convexity of the feasible domain of decision problem. Solutions are kept within feasible region during the mutation as well as crossover operations. Test problems are developed to evaluate the performance of the proposed algorithm and to determine satisficing decisions. In case of two objectives, weighting method is also applied to find the locus of optimal solutions.     

Nonlinear integer goal programming models for acceptance sampling

Two lexicographic goal programming models are developed for determining the optimal sample size and acceptance number for acceptance sampling plans in quality control. Both models address the conflicting criteria inherent in such sampling problems, namely the average lot inspection cost and the average outgoing quality. The first model assumes a known constant lot fraction defective, while the second relaxes this assumption and instead assumes knowledge of a prior distribution on the fraction of defectives. A three-phase algorithm is developed which exploits the problem structure in order to find optimal solutions after examining a small percentage of the feasible sampling plans. On a set of 64 test problems the algorithm always found the optimal solution, typically after evaluating only 3–5% (and never more than 9%) of the feasible points.     

Deadbeat control of linear discrete-time systems by periodic output-feedback

Two deadbeat control problems for open-loop, time-invariant linear, discrete-time multivariable systems by periodic output-feedbacks are formulated and solved. Sufficient conditions for the existence of solutions to the problems are established. Procedures for finding solutions to the problems are presented and illustrated by numerical examples.