Existing Optimization Methods Research Articles

Design Optimisation of Fully Actuated UAVs Using Hybrid Optimisation

During the past few years, there has been a rise in novel fully actuated multirotor UAV configurations, each customised to perform a different task. These configurations must be optimised to extract their full potential for the different use cases. The main issue with existing optimisation methods is the computational cost required to produce a design when the optimisation includes both continuous and discrete variables, such as off-the-shelf components. We propose a hybrid optimisation method using continuous surrogate methods with localised parameter sweep. The continuous stage aims at finding the optimal value for the continuous design variables and reduces the search space for the discrete design variables. Empirical models for the off-the-shelf components are used to reduce the size of the continuous stage of the optimisation. The localised search method consists of a parameter sweep of the discrete design variables within a certain threshold of the optimal parameters from the first stage. Three case studies confirm the method’s capabilities with different configurations and optimisation setups, comparing optimality, success rate and computational cost. The optimal design from the hybrid method is consistent with the baseline methods used for comparison within each case study, with a minimum success rate of 30% while reducing cost by 98%. Compared to specialised discrete methods, the improvement in computational cost is inconsistent; however, it achieves a reduction of 99.3% with certain design requirements. A comparison to another hybrid method was also performed, with the proposed method maintaining its cost, optimality and success rate better than the SQP-DSS method when increasing method complexity.

A Leadfield-Free Optimization Framework for Transcranially Applied Electric Currents.

Transcranial Electrical Stimulation (TES), Temporal Interference Stimulation (TIS), Electroconvulsive Therapy (ECT) and Tumor Treating Fields (TTFields) are based on the application of electric current patterns to the brain. The optimal electrode positions, shapes and alignments for generating a desired current pattern in the brain vary between persons due to anatomical variability. The aim is to develop a flexible and efficient computational approach to determine individually optimal montages based on electric field simulations. We propose a leadfield-free optimization framework that allows the electrodes to be placed freely on the head surface. It is designed for the optimization of montages with a low to moderate number of spatially extended electrodes or electrode arrays. Spatial overlaps are systematically prevented during optimization, enabling arbitrary electrode shapes and configurations. The approach supports maximizing the field intensity in target region-of-interests (ROI) and optimizing for a desired focality-intensity tradeoff. We demonstrate montage optimization for standard two-electrode TES, focal center-surround TES, TIS, ECT and TTFields. Comparisons against reference simulations are used to validate the performance of the algorithm. The system requirements are kept moderate, allowing the optimization to run on regular notebooks and promoting its use in basic and clinical research. The new framework complements existing optimization methods that require small electrodes, a predetermined discretization of the electrode positions on the scalp and work best for multi-channel systems. It strongly extends the possibilities to optimize electrode montages towards application-specific aims and supports researchers in discovering innovative stimulation schemes. The framework is available in SimNIBS.

Preload Multi-Objective Optimization Method for Ultrasonic Motors Based on NSGA-II

The preload has a significant impact on the output performance of ultrasonic motors. To achieve optimal output from the ultrasonic motor, this paper proposes a novel multi-objective optimization method for preload, utilizing the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to solve the problem of determining the optimal preload. Firstly, a platform for preload adjustment and testing is established to conduct experimental tests on the motor’s characteristics. Based on the experimental data, the influence of preload on the motor’s output performance is analyzed. Secondly, linear regression models are established for three performance indicators: no-load speed, stall torque, and output efficiency with respect to preload, and the objective functions are fitted accordingly. Finally, the NSGA-II algorithm is used for multi-objective optimization with three objectives to obtain the Pareto front and determine the optimal preload for the ultrasonic motor. Simulation and experimental results show that after preload optimization, the no-load speed increased by 4.93%, stall torque increased by 26.10%, and output efficiency increased by 11.28%. Compared to existing optimization methods, this approach has lower computational complexity and better optimization performance, ensuring higher optimization accuracy and precision. The preload optimization method based on NSGA-II can improve the performance of ultrasonic motors in practical applications.

Multi-Objective Topology Optimization of Thin-Plate Structures Based on the Stiffener Size and Layout

To address the limitations of existing optimization methods that focus on single objectives or neglect stiffener features, a multi-objective topology optimization (MOTO) method is proposed based on the stiffener size and layout. By constraining the initial structural performance parameters, the optimal stiffener height is determined through size optimization. Based on the stiffener height, single-objective topology optimization is used to achieve the best material distribution. The stiffener width is treated as a design variable, while MOTO is performed on the load point displacement, first natural frequency, and mass, thereby yielding an optimal stiffener width and performance. Finally, a multi-dimensional analysis of the stiffener height, width, and dynamic and static characteristics of the stiffened thin-plate structure is conducted. The results indicate that the optimized stiffener layout is considerably improved. Compared to the initial structure, the maximum and average displacements of the load point are reduced by 23.26% and 8.62%, respectively. The first natural frequency increases by 3.81%, while the maximum resonance amplitude and overall structural mass decrease by 39.97% and 1.99%, respectively. The results indicate that the optimized structure achieves a lightweight design while maintaining better stiffness and low-frequency vibration resistance. The feasibility and effectiveness of the proposed method are validated.

Static voltage bilayer optimization for distribution networks based on Sobol’ method

Abstract In active distribution networks, the uncertainty of a large number of distributed power sources makes the voltage stability problem of distribution networks more and more prominent. The existing optimization methods tend to deal with the economy and voltage stability of distribution networks as independent objectives, with less consideration of their connection, making it difficult to realize the efficient use of flexible resources. The paper introduces upper and lower connection factors and proposes a two-layer optimization model for distribution network voltage using the Sobol’ method. First, taking into account the uncertainty of distributed power sources and loads in active distribution networks, a probabilistic tidal current calculation model for active distribution networks is constructed, based on which the impact of load fluctuations on the L index is quantitatively analyzed by the global sensitivity of the Sobol’ method. Secondly, the upper and lower connection factors are calculated based on the sensitivity analysis results. The upper economic objective and the lower stability objective are connected through the connection factors to establish a two-layer optimization model to solve the flexible resources in the system optimally. Finally, simulations on the modified IEEE33 node system validate that the proposed optimization model enhances distribution network security while considering economic factors.

Deep Neural Networks for Estimating Regularization Parameter in Sparse Time–Frequency Reconstruction

Time–frequency distributions (TFDs) are crucial for analyzing non-stationary signals. Compressive sensing (CS) in the ambiguity domain offers an approach for TFD reconstruction with high performance, but selecting the optimal regularization parameter for various signals remains challenging. Traditional methods for parameter selection, including manual and experimental approaches, as well as existing optimization procedures, can be imprecise and time-consuming. This study introduces a novel approach using deep neural networks (DNNs) to predict regularization parameters based on Wigner–Ville distributions (WVDs). The proposed DNN is trained on a comprehensive dataset of synthetic signals featuring multiple linear and quadratic frequency-modulated components, with variations in component amplitudes and random positions, ensuring wide applicability and robustness. By utilizing DNNs, end-users need only provide the signal’s WVD, eliminating the need for manual parameter selection and lengthy optimization procedures. Comparisons between the reconstructed TFDs using the proposed DNN-based approach and existing optimization methods highlight significant improvements in both reconstruction performance and execution time. The effectiveness of this methodology is validated on noisy synthetic and real-world signals, emphasizing the potential of DNNs to automate regularization parameter determination for CS-based TFD reconstruction in diverse signal environments.

Improved state transition algorithm for hybrid power system reactive power optimization using filter techniques

Abstract Reactive power optimization in power systems is crucial for their normal and efficient operation. However, with the increasing prevalence of active current and direct current (AC-DC) hybrid power grids, existing optimization methods face issues such as poor adaptability, low efficiency, insufficient accuracy, and inadequate convergence. To address these problems, this paper proposes a reactive power optimization model based on an improved state transition algorithm (STA) utilizing filter techniques. The optimization model considers operational costs associated with adjusting transformer tap positions, switching reactive power compensation capacitors, regulating generator terminal voltages, and adjusting active power in DC transmission. Simulation results demonstrate that this method can reduce system losses and voltage control costs while ensuring stable system operation, thus improving the reactive power distribution in AC-DC hybrid transmission systems.

Deep reinforcement learning-based multi-objective optimization for electricity–gas–heat integrated energy systems

With the increasing global attention on energy efficiency and carbon emissions, the optimization of integrated energy systems (IES) has become the key to improve energy efficiency and reduce pollution emissions. However, most of the existing optimization methods cannot effectively deal with the complexity of high dimensional continuous action space. Therefore, this paper focuses on a novel multi-objective optimization strategy for the electricity–gas–heat integrated energy systems (EGH-IES). Firstly, considering the absorption capacity of wind power and the emission of pollutant gases, a multi-objective optimization model is constructed based on the mechanism model and operation constraints of each device in EGH-IES, in which the integrated operation cost and the environmental factors are taken as optimization objectives. Then, the multi-objective optimization problem is designed as the optimal strategy of interaction learning between agent and environment in reinforcement learning, and the output power of the devices constitutes the action of reinforcement learning. Additionally, the Ornstein–Uhlenbeck process is introduced to enhance the training efficiency and exploration performance of the agent, and the deep deterministic policy gradients (DDPG) algorithm is employed to optimize the action, thus the output power of the appliances could be obtained. Finally, the simulation results show that compared with deep Q network (DQN) method and proximal policy optimization (PPO) method, the reward function value of the proposed method increases by 2.43% and 6.09%, respectively, which represents a reduction in economic cost and pollutant emissions. These verify the effectiveness and superiority of the proposed multi-objective optimization scheme in cost reduction and benefit improvement for the EGH-IES.

Optimal rule-based energy management and sizing of a grid-connected renewable energy microgrid with hybrid storage using Levy Flight Algorithm

The study addresses the integration of hybrid hydrogen (H2) and battery (BT) energy storage systems into a renewable energy microgrid comprising solar photovoltaic (PV) and wind turbine (WT) systems. The research problem focuses on improving the effectiveness and computational efficiency of energy management systems (EMS) while ensuring high system reliability. Despite the existing optimization methods for hybrid microgrids, challenges remain in optimizing energy storage and capacity planning in grid-connected microgrids. To solve this, we propose the use of the Levy Flight Algorithm (LFA) to optimize the capacities of PV, WT, H2 tanks, electrolyzers (EL), fuel cells (FC), and BT, which presents a complex nonlinear optimization challenge. The novelty of this study lies in integrating the LFA with a rule-based EMS, enhancing system reliability and efficiency. The proposed approach significantly reduces the annualized system cost (ASC) and the levelized cost of energy (LCOE). The result demonstrate that the LFA outperforms methods like the Salp Swarm Algorithm (SSA), Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO) and Genetic Algorithm (GA), yielding cost savings of $3,309, $5,297, $4,484, and $5,129 respectively. The LFA achieves the lowest LCOE at $0.275/kWh, compared to $0.278/kWh with SSA, $0.289/kWh with GA, $0.280/kWh with PSO and $0.283/kWh with GWO. This research contributes to the broader scientific community by providing a more efficient approach to optimizing renewable energy microgrids with hybrid storage systems, thus promoting eco-friendly and cost-effective energy solutions. The proposed system design offers a pathway to future energy systems with high renewable integration, especially as technology advances and costs continue to decrease.

Salmon Salar Optimization: A Novel Natural Inspired Metaheuristic Method for Deep-Sea Probe Design for Unconventional Subsea Oil Wells

As global energy demands continue to rise, the development of unconventional oil resources has become a critical priority. However, the complexity and high dimensionality of these problems often cause existing optimization methods to get trapped in local optima when designing key tools, such as deep-sea probes. To address this challenge, this study proposes a novel meta-heuristic approach—the Salmon Salar Optimization algorithm, which simulates the social structure and collective behavior of salmon to perform high-precision searches in high-dimensional spaces. The Salmon Salar Optimization algorithm demonstrated superior performance across two benchmark function sets and successfully solved the constrained optimization problem in deep-sea probe design. These results indicate that the proposed method is highly effective in meeting the optimization needs of complex engineering systems, particularly in the design optimization of deep-sea probes for unconventional oil exploration.

Optimizing energy expenditure in agricultural autonomous ground vehicles through a GPU-accelerated particle swarm optimization-artificial neural network framework

The accurate energy consumption prediction in Agricultural Ground Vehicles (AGVs) holds immense potential for optimizing operational efficiency and minimizing environmental impact. However, existing optimization methods for such prediction tasks often suffer from high computational demands, hindering their practical implementation. This paper introduces a ground-breaking approach that overcomes this limitation by leveraging the potent computational power of Graphics Processing Units (GPUs) to accelerate the optimization process dramatically. We propose a novel adaptation of the Particle Swarm Optimization (PSO) algorithm, specifically tailored to the intricate multi-objective challenges of AGV energy prediction. This framework harnesses the strengths of a multi-objective approach, enabling the simultaneous optimization of prediction accuracy and model complexity. To further enhance efficiency, we seamlessly integrate GPU parallelization techniques, significantly expediting both the optimization process and the training of Artificial Neural Networks (ANNs) employed for prediction. Preliminary results demonstrate a remarkable improvement in the accuracy of AGV energy consumption predictions, directly attributed to the synergistic effect of optimizing the ANN architecture and parameters through our proposed PSO framework. This tailored PSO adaptation distinguishes itself by its ability to tackle the complex multi-objective nature of AGV energy prediction with enhanced efficiency and precision. It thus emerges as a compelling and novel solution within the realm of Machine Learning and heuristic methods for agricultural robotics, paving the way for sustainable and optimal AGV operations.

Agile Optimization Framework: A framework for tensor operator optimization in neural network

In recent years, with the gradual slowing of Moore’s Law and the development of deep learning, the demand for hardware performance of executing deep learning based applications has significantly increased. In this case, deep learning compilers have been proven to maximize hardware performance while keeping computational power constant, especially the end-to-end compiler Tensor Virtual Machine (TVM). TVM optimizes tensors by finding excellent parallel computing schemes, thereby achieving the goal of improving the performance of neural network inference. However, there is still untapped potential in current optimization methods. However, existing optimization methods based on the TVM, such as Genetic Algorithms Tuner (GA-Tuner), have failed to achieve a balance between optimization performance and optimization time. The intolerable duration of optimization detracts from TVM’s usability, rendering it challenging to extend into the scientific community. This paper introduces a novel deep learning compilation optimization framework base on TVM called Agile Optimization Framework (AOF), which incorporates a tuner based on the latest Beluga Whale Optimization Algorithm (BWO). The BWO is adept at tackling complex problems characterized by numerous local optima, making it particularly suitable for hardware compilation optimization scenarios. We further propose an Evolving Epsilon Strategy (EES), a search strategy that adaptively adjusts the balance between exploration and exploitation, thereby enhancing the effectiveness of the algorithm. Additionally, we developed a supervised Tuning Accelerator (TA) aimed at reducing the time required for optimization and enhancing efficiency. Comparative experiments demonstrate that AOF achieves 11.36%–66.20% improvement in performance and 30.30%–54.60% reduction in optimization time, significantly outperforming the control group.

Control system for automatic adjustment of departure intervals of public transport vehicles

In numerous urban areas, fine-tuning bus schedules to align with fluctuating passenger numbers poses a sizable challenge, compounded by diverse factors such as weather and unexpected events. This paper conducts a comprehensive examination of existing optimization methods aimed at enhancing the departure intervals of public transport systems, particularly focusing on the Bidirectional Coordination Optimization Algorithm (BCOA) and the genetic algorithm. Through systematic analysis, these methodologies are meticulously explored for their effectiveness in addressing the scheduling challenges faced by urban transit systems. Additionally, it delves into specific instances where the Proportional-Integral-Derivative (PID) control mechanism has been employed to adeptly adjust the timing of bus departures based on real-time demand. The PID system's proven reliability and efficacy spotlight its potential as a transformative tool for advancing public transit systems. By leveraging such technology, there's tangible optimism for achieving a more responsive, efficient, and passenger-friendly public transportation network in the years to come.

Clothing-invariant contrastive learning for unsupervised person re-identification

Clothing change person re-identification (CC-ReID) aims to match images of the same person wearing different clothes across diverse scenes. Leveraging biological features or clothing labels, existing CC-ReID methods have demonstrated promising performance. However, current research primarily focuses on supervised CC-ReID methods, which require a substantial number of manually annotated labels. To tackle this challenge, we propose a novel clothing-invariant contrastive learning (CICL) framework for unsupervised CC-ReID task. Firstly, to obtain clothing change positive pairs at a low computational cost, we propose a random clothing augmentation (RCA) method. RCA initially partitions clothing regions based on parsing images, then applies random augmentation to different clothing regions, ultimately generating clothing change positive pairs to facilitate clothing-invariant learning. Secondly, to generate pseudo-labels strongly correlated with identity in an unsupervised manner, we design semantic fusion clustering (SFC), which enhances identity-related information through semantic fusion. Additionally, we develop a semantic alignment contrastive loss (SAC loss) to encourages the model to learn features strongly correlated with identity and enhances the model’s robustness to clothing changes. Unlike existing optimization methods that forcibly bring closer clusters with different pseudo-labels, SAC loss aligns the clustering results of real image features with those generated by SFC, forming a mutually reinforcing scheme with SFC. Experimental results on multiple CC-ReID datasets demonstrate that the proposed CICL not only outperforms existing unsupervised methods but can even achieves competitive performance with supervised CC-ReID methods. Code is made available at https://github.com/zqpang/CICL.

Expert knowledge data-driven based actor–critic reinforcement learning framework to solve computationally expensive unit commitment problems with uncertain wind energy

With the expansion of power grid, unaffordable computational cost and time will pose serious challenges of time-efficient scheduling in unit commitment problem (UCP). However, existing optimization methods, i.e., mathematical programming methods and meta-heuristic algorithms, are powerless and time-consuming to handle computationally expensive UCP (CEUCP). Thus, reinforcement learning methods with strong inference and time-saving performances are motivated to solve the computationally expensive challenges in tackling CEUCPs. In this paper, a novel expert knowledge data-driven based actor–critic (AC) reinforcement learning methodology is proposed for solving CEUCPs. Specifically, in the proposed AC reinforcement learning methodology, expert knowledge, data-driven surrogate model, and improved meta-heuristic algorithm are integrated for further performance enhancement. Firstly, a novel action selection mechanism (based on the expert knowledge of thermal units characteristic) is integrated into AC to improve the efficiency of network training. Secondly, an improved extreme learning machine (ELM) data-driven surrogate model is proposed to build reward function in AC. In detail, original cost function in reward is replaced by a lightweight ELM model. Shape distance is integrated into ELM for enhancing accuracy. Finally, original marine predators algorithm (MPA) is improved for obtaining optimal dispatching decisions and rewards of AC method quickly and correctly. Original search pattern is replaced by quantum based representation for boosting convergence. The excellent performances of the proposed AC framework are verified by simulations of 10-units, 100-units, and 100-units with wind energy test systems.

An Adaptive Dual-Mode Task-Oriented Resource Management Strategy for GEO Relay Systems

With the fierce global competition on satellite networks, the building of satellite constellations grows explosively. Sharply increasing on-orbit data will face the challenge of satellite-ground data transmission. GEO satellites become the top choice for satellite data relay due to their stable satellite-ground link. Most existing spectrum resource management for GEO relays is equipment-oriented and benefit priority, which may lead to a waste of spectrum resources. In this paper, we propose a real-time task-oriented resource allocation strategy for GEO relay systems. We model the spectrum allocation problem as a distributed non-cooperative Stackelberg game process. We prove that when both sides of the game pursue the maximization of personal revenue, the system will enter a Nash equilibrium state, whereas spectrum resources are not fully used. Based on the maximization of individual utilities (U-prior) and spectrum utilization (S-prior) methods, we design an adaptive dual-mode pricing mode to maximize the spectrum resources within a certain loss of revenue. The simulation results show that the S-prior and U-prior have better performance than the baseline method and existing optimization methods. Our proposed dual-mode strategy is making more throughputs and has less delay with little loss of utility values than that of individual utility maximization.

Joint cooperative caching and power control for UAV-assisted internet of vehicles

In view of the current problems of spectrum resource scarcity, return congestion, and insufficient energy utilization in the unmanned aerial vehicle (UAV)-assisted Internet of Vehicles (IoV), this paper investigates the cooperative caching and power control, and proposes a joint optimization method to improve the overall Energy Efficiency (EE) . In this method, we first propose a communication establishment threshold to control the V2V communication distance and serve as a joint optimization factor. Then we derive the closed form expressions of offloading ratio and EE of the UAV-assisted IoV, and formulate the optimization problem of maximizing EE. Due to the coupling relationship between caching strategy and transmission power, it is difficult for us to directly solve the optimization problem. Furthermore, we propose an alternating optimization algorithm for solving the optimization problem. Finally, the experimental simulation compare the propose joint optimization method with other existing optimization methods, and the simulation results prove the effectiveness and superiority of the propose joint optimization method.

A practical optimization framework for political redistricting: A case study in Arizona

Following the decennial census, each state in the U.S. redraws its congressional and state legislative district boundaries, which must satisfy various legal criteria. For example, Arizona’s Constitution describes six legal criteria, including contiguity, population balance, competitiveness, compactness, and the preservation of communities of interest, political subdivisions and majority–minority districts, each of which is to be enforced “to the extent practicable”. Optimization algorithms are well suited to draw district maps, although existing models and methods have limitations that inhibit their ability to draw legally-valid maps. Adapting existing optimization methods presents two major challenges: the complexity of modeling to achieve multiple and subjective criteria, and the computational intractability when dealing with large redistricting input graphs. In this paper, we present a multi-stage optimization framework tailored to redistricting in Arizona. This framework combines key features from existing methods, such as a multilevel algorithm that reduces graph input sizes and a larger local search neighborhood that encourages faster exploration of the solution space. This framework heuristically optimizes geographical compactness and political competitiveness while ensuring that other criteria in Arizona’s Constitution are satisfied relative to existing norms. Compared to Arizona’s enacted map (CD118) to be used until 2032, the most compact map produced by the algorithm is 41% more compact, and the most competitive map has five more competitive districts. To enable accessibility and to promote future research, we have created Optimap, a publicly accessible tool to interact with a part of this framework. Beyond the creation of these maps, this case study demonstrates the positive impact of adapting optimization-based methodologies for political redistricting in practice.

Improving classification accuracy of fine-tuned CNN models: Impact of hyperparameter optimization

The immense popularity of convolutional neural network (CNN) models has sparked a growing interest in optimizing their hyperparameters. Discovering the ideal values for hyperparameters to achieve optimal CNN training is a complex and time-consuming task, often requiring repetitive numerical experiments. As a result, significant attention is currently being devoted to developing methods aimed at tailoring hyperparameters for specific CNN models and classification tasks. While existing optimization methods often yield favorable image classification results, they do not provide guidance on which hyperparameters are worth optimizing, the appropriate value ranges for those hyperparameters, or whether it is reasonable to use a subset of training data for the optimization process. This work is focused on the optimization of hyperparameters during transfer learning, with the goal of investigating how different optimization methods and hyperparameter selections impact the performance of fine-tuned models. In our experiments, we assessed the importance of various hyperparameters and identified the ranges within which optimal CNN training can be achieved. Additionally, we compared four hyperparameter optimization methods—grid search, random search, Bayesian optimization, and the Asynchronous Successive Halving Algorithm (ASHA). We also explored the feasibility of fine-tuning hyperparameters using a subset of the training data. By optimizing the hyperparameters, we observed an improvement in CNN classification accuracy of up to 6%. Furthermore, we found that achieving a balance in class distribution within the subset of data used for parameter optimization is crucial in establishing the optimal set of hyperparameters for CNN training. The results we obtained demonstrate that hyperparameter optimization is highly dependent on the specific task and dataset at hand.

Synthesis of ship systems optimal regulators based on matrix inequalities

The output and state regulators synthesis for ship control systems based on linear matrix inequalities using the CVX software package for solving optimization problems by the method of semi-definite programming is considered. Estimates parameters for regulators providing asymptotically stable systems regimes on Lyapunov inequalities solutions sets are obtained. An algorithm for controlling the ship course using a robust regulator in state feedback, followed by the use of CVX technologies to implement the minimum energy consumption control algorithm, is presented. The control vector estimation is reduced to the minimization of the L2 norm and it is performed in two stages, namely, the best mode is first determined, and then the mode under constraints is defined. The use of Lyapunov inequalities with the calculations implementation in CVX format on proprietary solvers has allowed us to obtain compact program texts and expand the set of restrictions introduced. In the algorithm for solving the boundary value problem of controlling a discrete model of an object, in contrast to existing optimization methods, a mode of transfer from the initial state to the final one with the right boundary, not necessarily equal to zero, is implemented. For a class of observable and controllable time–invariant systems, using the Krylov’s function contained in the MATLAB gallery of functions, L2‑norms to be minimized are formed. Examples of regulator calculations with a given degree of stability and optimal systems, confirming proposed solutions correctness, are given.