Efficiency in uncertain variational control problems

Supervisory Event-Triggered Control of Uncertain Process Networks: Balancing Stability and Performance

This work presents a methodological framework for the design of a resource-aware supervisory control system for process networks with model uncertainty and communication resource constraints. The developed framework aims to balance the objective of closed-loop stabilization of the overall network with that of meeting the local performance requirements of the component subsystems while keeping the rate of data transfer between the local control systems to a minimum. First, a quasi-decentralized networked control structure, with a set of local model-based controllers communicating with one another over a shared communication medium at discrete times, is designed. A Lyapunov stability analysis of the closed-loop system is then carried out, and the results are used to derive appropriate bounds on the local model state estimation errors as well as the dissipation rates of the local control Lyapunov functions. These bounds are used as stability and performance thresholds to trigger communication between the local control systems and a higher-level supervisor that coordinates the transfer of state measurements between the distributed control systems. A breach of the local stability and performance thresholds generates alarm signals which are transmitted to the supervisor to determine which subsystems should communicate with one another. The supervisor employs a composite Lyapunov function to assess the impact of the local threshold breaches on the stability of the overall closed-loop system. The supervisory communication logic takes account of the evolution of the local and composite Lyapunov functions in order to balance the stability and local performance requirements. Finally, the developed framework is demonstrated using a representative chemical process network and compared with other unsupervised event-based control approaches. It is shown that the supervisory event-based control approach leads to a more judicious utilization of network resources that helps improve closed-loop process performance in the presence of unexpected disturbances and input rate constraints.

Optimality Conditions and Duality for a Class of Generalized Convex Interval-Valued Optimization Problems

This paper is devoted to derive optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrically derivative. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend above optimization conditions. Examples are also presented to illustrate corresponding results.

On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution.

Second-Order PDE Constrained Controlled Optimization Problems with Application in Mechanics

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.

A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

On a Dual Pair of Multiobjective Interval-Valued Variational Control Problems

In this paper, by using the new concept of (ϱ,ψ,ω)-quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under (ϱ,ψ,ω)-quasiinvexity hypotheses for the considered class of optimization problems.

Assessing a Multi-Objective Genetic Algorithm with a Simulated Environment for Energy-Saving of Air Conditioning Systems with User Preferences

Electricity is one of the most important resources for the growth and sustainability of the population. This paper assesses the energy consumption and user satisfaction of a simulated air conditioning system controlled with two different optimization algorithms. The algorithms are a genetic algorithm (GA), implemented from the state of the art, and a non-dominated sorting genetic algorithm II (NSGA II) proposed in this paper; these algorithms control an air conditioning system considering user preferences. It is worth noting that we made several modifications to the objective function’s definition to make it more robust. The energy-saving optimization is essential to reduce CO2 emissions and economic costs; on the other hand, it is desirable for the user to feel comfortable, yet it will entail a higher energy consumption. Thus, we integrate user preferences with energy-saving on a single weighted function and a Pareto bi-objective problem to increase user satisfaction and decrease electrical energy consumption. To assess the experimentation, we constructed a simulator by training a backpropagation neural network with real data from a laboratory’s air conditioning system. According to the results, we conclude that NSGA II provides better results than the state of the art (GA) regarding user preferences and energy-saving.

On Modified Interval-Valued Variational Control Problems with First-Order PDE Constraints

In this paper, a modified interval-valued variational control problem involving first-order partial differential equations (PDEs) and inequality constraints is investigated. Specifically, under some generalized convexity assumptions, we formulate and prove LU-optimality conditions for the considered interval-valued variational control problem. In order to illustrate the main results and their effectiveness, an application is provided.